STODDARD      SERIES. 


GENERAL    GEOMETRY 


AND 


CALCULUS'. 


INCLUDING  BOOK  L    OF  THE   GENERAL   GEOMETRY,    TREATING   OP   LOCI  EH 
A   PLANE  J    AND   AN   ELEMENTARY    COURSE    IN   THE   DIFFER 
ENTIAL   AND    INTEGRAL   CALCULUS. 


BT 


EDWAKD  OLNEY, 


WOFBSSOB  OF  MATHEMATICS  IN  THK   UNIVERSITY  OF  MICHIOAH. 


NEW  YORK : 
SHELDON     AND     COMPANY 


Entered  according  to  Act  of  Congress  in  the  year  1871,  by 

SHELDON   &   COMPANY, 
In  the  Office  of  the  Librarian  of  Congress  at  Washington. 


Stoddard's  Mathematical  Series. 

itoddard's  Juvenile  Mental  Arithmetic  .............................  $    25 

itoddard's  Intellectual  Arithmetic  ..................................       40 

itoddard's  Rudiments  of  Arithmetic  ...............  .................       50 

Itoddard'a  New  Practical  Arithmetic  ..............................  1  00 

Short  and  Full  Course  for  Graded  Schools. 

ftoddard's  Pictorial  Primary  Arithmetic  ............................       30 

fcoddard's  Combination  Arithmetic  .................................       75 

itoddard's  Complete  Arithmetic  ....................................  1  25 

The  Combination  School  Arithmetic,  being  Mental  and  Written  Arithmetic  in 
ne  boqfe,  will  alone  serve  for  District  Schools.  For  Academics  a  full  high  course 
5  obtained  by  the  Complete  Arithmetic  and  Intellectual  Arithmetic. 

Stoddard's  Higher  Mathematics. 

L  Complete  School  Algebra,  in  one  vol.,  390  pages.  $1  50.  Designed  for  Ele 
mentary  and  Higher  Classes  in  Schools  and  Academies.  By  Prof.  EDWABD  OLNEY, 
University  of  Michigan, 

L  Geometry  and  Trigonometry,  in  one  vol.     By^Prof.  EDWARD  OLNEY. 

L  General  Geometry  anoV  Galculus/.i'n  pitfe  V&1.  1 


,%    The   other  bookof'Sftp^dardyiSeJdesr  w\ll  'D&  published  as  rapidly  as 
ossible.  -.'.'.•     •  •  .  j  J  /•  *.  •  e  / 


PREFACE. 


THIS  volume  presents  a  course  in  the  General  Geometry  an< 
the  Infinitesimal  Calculus,  which  is  thought  to  be  as  extended  a 
is  practicable  for  the  general  student  in  the  regular  undergrad 
uate  course  in  our  American  colleges.  If  we  can  secure  a  suffi 
ciently  high  grade  of  preparation,  so  that  students  in  the  Fresh 
man  year  can  complete  a  respectable  course  in  Elementar 
Geometry,  including  Plane  and  Spherical  Trigonometry,  and  i] 
Algebra,  it  is  thought  that  during  the  Sophomore  year  the  con 
tents  of  this  volume  can  be  readily  mastered.  Such  is  th 
purpose  in  this  University ;  and  it  is  already  well  nigh  realized. 

As  to  the  propriety  of  including  the  study  of  both  these  sub 
jects  in  the  regular  undergraduate  course,  there  can  be  but  on 
opinion  among  those  competent  to  judge.  No  man  can  justl; 
claim  to  have  a  good  general  education,  who  is  ignorant  of  th 
elements  of  the  processes  by  which  all  extended  operations  i] 
the  exact  sciences  are  carried  forward,  and  which  are  the  foun 
dation  of  all  the  arts  based  upon  mathematical  science.  Th 
man  who  is  ignorant  of  the  General  Geometry  and  the  Calculus 
is  not  only  a  stranger  in  one  of  the  sublimest  realms  of  huma] 
thought,  but  knows  nothing  of  the  instruments  in  most  familia 
use  by  the  engineer,  the  astronomer,  and  the  machinist  in  any  o 
the  higher  walks  of  art.  In  short  he  is  ignorant  of  the  charac 
teristic  processes  of  the  mathematician  of  his  day. 

Nor  is  it  impracticable  for  the  majority  of  students  to  becom< 
intelligent  in  these  subjects.  They  do  not  lie  beyond  the  read 
of  good  common  minds,  nor  require  peculiar  mental  character 
istics  for  their  mastery.  The  difficulty  hitherto  has  been  in  th< 
methods  of  presentation,  in  the  limited  and  totally  inadequate 
amount  of  time  assigned  them,  and  more  than  all  in  the  precon 
ceived  notion  of  their  abstruseness. 

The  mathematician  will  see  in  the  plan  of   the  first  part  o: 


IV  PREFACE. 

this  volume,  as  well  as  in  its  title  ( General  Geometry),  a  recog 
nition  of  the  profound  views  of  Comte  upon  the  philosophy  of 
the  science.  This  science  is  a  method  of  Geometrical  reasoning. 
Its  characteristic  feature  is  that  it  represents  form,  as  well  as 
magnitude,  by  equations,  and  hence  makes  algebra  its  instru 
ment.  It  is  consequently  indirect.  Its  ultimate  object  is  breadth 
of  comprehension, — the  discussion  of  general  problems.  In 
accordance  with  this  conception,  the  first  purpose  is  to  exhibit 
the  method  of  translating  geometrical  forms  into  algebraic 
equations,  i.  e.  to  show  how  loci  are  represented  by  equations. 
While  the  prominence  is  given  to  the  Conic  Sections  which 
their  importance  in  physical  science  demands,  the  student  is 
not  led  to  think  that  this  is  merely  a  scheme  for  treating  these 
curves.  He  is  taught  to  look  upon  it  as  a  method  of  investiga 
tion — as  designed  to  embrace  the  discussion  of  all  loci.  For  this 
purpose  many  Higher  Plane  Curves  are  treated.  After  the 
student  has  become  familiar  with  the  equation  as  the  represent 
ative  of  a  locus,  and  has  learned  how  to  produce  the  equation  of 
a  locus  from  its  definition,  he  has  obtained  the  instrument.  He 
is  now  to  learn  how  to  apply  it  for  the  purposes  of  Geometrical 
investigation.  In  carrying  forward  this  part  of  his  study  the 
Calculus  renders  invaluable  service.  Moreover,  this  preparatory 
study  of  the  General  Geometry  gives  him  exactly  the  needed 
means  for  illustrating  the  elementary  processes  of  the  Calculus. 
He  has,  therefore,  come  to  a  point  where  his  further  progress 
requires  a  knowledge  of  the  Differential  Calculus,  and  he  has 
also  the  requisite  preparation  for  its  study.  Hence,  after  having 
become  familiar  with  the  first  three  chapters  of  General  Geom 
etry,  he  reads  the  Differential  Calculus. 

By  this  arrangement  the  Calculus  is  seen  in  its  true  relations, 
as  an  independent  abstract  science,  grand  and  beautiful  in  itself, 
and  rendering  most  efficient  service  in  the  more  immediately 
practical  science  of  Geometry,  as  it  is  afterwards  seen  to  do  in 
Physics.  Having  obtained  the  needed  acquaintance  with  the 
Differential  Calculus,  the  student  returns  to  pursue  his  Geomet 
rical  studies,  with  equations  of  loci  as  his  instruments,  and  the 
Calculus  to  aid  in  the  manipulation  of  them.  But  the  peculiar 
features  of  the  treatise  are  too  numerous  to  be  enumerated  here, 
and  can  only  be  seen  in  their  true  light  by  a  perusal  of  the 
work. 


PREFACE.  V 

In  the  treatment  of  the  Calculus  I  have  used  the  Infinitesi 
mal  method  instead  of  the  method  of  Limits,  on  account  of  its 
greater  simplicity,  as  well  as  because  it  is  the  only  conception 
which  enables  us  to  apply  the  Calculus  to  practical  problems 
with  any  degree  of  facility.  The  general  use  of  the  method  of 
limits  in  our  text  books  has  done  not  a  little  to  prevent  the  com 
mon  study  of  this  elegant  and  useful  branch  of  mathematics. 
This  method  is  not  only  exceedingly  cumbrous,  but  it  has  the 
misfortune  that  its  element,  a  differential  coefficient,  is  a  ratio. 
The  abstract  nature  of  a  ratio,  and  the  fact  that  it  is  a  com 
pound  concept,  peculiarly  unfit  it  for  elementary  purposes.  The 
beginner  will  never  use  it  with  satisfaction,  for  it  does  not  give 
him  simple,  direct  and  clearly  defined  conceptions.  But  while  I 
have  adopted  the  infinitesimal  theory,  I  have  felt  free  to  intro 
duce  the  doctrine  of  limits,  and  to  illustrate  and  apply  it.  The 
metaphysical  objections  to  this  method,  if  not  rebutted  by  equal 
difficulties  of  a  similar  character  encountered  in  the  method  of 
limits,  are  immensely  overborne  by  its  practical  advantages ; 
for,  let  it  be  remembered  that  no  writer  adheres  to  the  Newton 
ian  method  throughout,  but  glides  into  the  other  in  the  Integral 
Calculus,  and  adopts  it  exclusively  in  most  geometrical  and 
physical  applications. 

The  sources  from  which  the  material  has  been  drawn  will  be 
readily  perceived  by  the  mathematician,  and  need  not  be  enum 
erated  here.  That  the  treatise  is  sufficiently  different  from 
others  of  a  similar  purpose  to  justify  its  existence,  the  author 
feels  more  sure  than  that  these  differences  will  commend  them 
selves  to  his  fellow  laborers  in  the  work  of  mathematical  train 
ing.  One  thing,  however,  is  certain,  nothing  in  matter,  arrange 
ment,  or  manner  of  treatment,  has  been  introduced  without 
careful  reference  to  the  capabilities  and  wants  of  such  students 
as  I  have  been  accustomed  to  meet  in  the  class  room  for  more 
than  twenty  years ;  and  few  things  will  be  found  in  the  volume 
but  what  have  been  put  to  the  test  of  class  room  use  many 
times  over. 

A  second  volume,  treating  of  Loci  in  Space,  and  affording  a 
more  extended  course  in  the  Calculus,  will  be  published  as  soon 
as  it  can  be  prepared.  The  present  is  thought  sufficient  for  all 
students  except  such  as  make  mathematics  a  specialty  ;  and  for 
the  latter  the  other  volume  will  be  designed. 


VI  PREFACE. 

In  conclusion  I  must  do  myself  the  pleasure  to  acknowledge 
my  indebtedness  to  my  accomplished  colleague  and  friend,  Prof. 
J.  C.  Watson,  Ph.  D.,  for  the  original,  direct,  and  simple  method 
of  demonstrating  the  rule  for  differentiating  a  logarithm,  which 
is  given  on  page  25,  and  which  banishes  from  the  Calculus  the 
last  necessity  for  resort  to  series  to  establish  any  of  its  funda 
mental  operations.  I  am  also  indebted  to  my  friend  and  pupil, 
J.  B.  Webb,  B.  S.,  for  many  valuable  suggestions,  and  much  care 
ful  labor  in  reading  both  the  manuscript  and  proof.  To  his 
quick  and  accurate  eye,  and  his  good  taste  and  logical  acumen, 
I  am  indebted  for  the  elimination  of  not  a  few  defects  which 
might  otherwise  have  disfigured  the  work.  That  there  is  not 
much  of  the  same  sort  of  pruning  yet  needed,  I  have  not  the 
vanity  to  think.  But,  such  as  it  is,  I  commend  my  work  to  the 
consideration  of  teacher  and  student,  with  the  hope  that  it  may 
contribute  to  aid  the  one  in  imparting,  and  the  other  in  acquiring, 
a  knowledge  of  the  elements  of  two  branches  of  science  which, 
in  their  fuller  developments,  exhibit  the  profoundest  and  most 
sagacious  workings  of  the  human  mind,  and  reach  to  the  farthest 
verge  of  the  hitherto  explored  realms  of  human  thought. 

EDWABD   OLNEY. 

ANN  AUBOK,  Mich.,  July,  1871. 


N.  B. — A  shorter  course  in  the  General  Geometry,  without  the 
Calculus,  may  be  taken  from  this  volume  lyy  such  as  desire  it. 
For  this  purpose,  the  first  three  chapters  are  to  be  read,  and  then 
the  course  completed  ~by  reading  the  XIV.  and  XV.  Sections  of  Chap 
ter  IV.  If  time  and  purpose  permit,  Articles  (194, 195)  might  be 
read  with  profit  by  such  students.  This  will  be  found  to  comprise  a 
course  on  Plane  Co-ordinate  Geometry  somewhat  more  full  than  is 
found  in  our  common  text-books. 


CONTENTS. 


INT11OD  UCTION. 

A  BKIEF  SURVEY  OF  THE  OBJECTS  OF  PURE  MATHEMATICS  AND  OF 
THE  SEVERAL  BRANCHES. 

PAGE 

PURE  MATHEMATICS.— Definition  (1)  ;    Branches  enumerated  (2,3} 1 

QUANTITY.  —  Definition  (4=} '.     1 

NUMBER. — Definition  (5)  ;    Discontinuous  and  continuous  (/>_,  7,8) 2,  3 

DEFINITION    or   THE  SEVERAL  BRANCHES    or  MATHEMATICS. — Arithmetic    (9;  ; 

Algebra  (1O);  Calculus  (11);  Geometry  (12)',  Descriptive  Geometry  (13)  3,  5 
GENERAL  GEOMETRY  divided  into  Two  Books  (14:) ^ . . . .     5 


GENERAL  GEOMETRY. 

BOOK  I. 

OF    PLANE    LOCI. 


CHAPTEK  I. 
THE   CARTESIAN   METHOD    OF   CO-ORDINATES. 


SECTION  I. 

DEFINITIONS  AND  FUNDAMENTAL  NOTIONS. 

Locus. — Definition  (1) 6 

GENERAL  GEOMETRY.  —Definition  (2) 6 

METHOD  OF  CO-ORDINATES. — What  (3)  ;    Two  Systems  (4)  ;    Varieties  of  Kec- 

tilinear  (5) 6,  7 

DEFINITIONS.— Axes  (6,  7)  ;    Origin  (8)  ;    Co-ordinates  (9,  W,  11)  ;    Illus 
tration  - 7,  8 

NOTATION.— Of   Co-ordinates   (12)  ;    The  Four  Angles  (13)  ;    Signs   of  the 

Co-ordinates  (14^ " 8'  9 

QUANTITIES.— Constant   and  Variable  (15)  ;    Definition  of  each  (16,17); 
Illustration  ;    Caution.     Sch.  1 , • 


Vlll  CONTENTS. 

PAGE 

INDETERMINATE  ANALYSIS. — What,  Sch.  2 9 

To  CONSTRUCT  AN  EQUATION. — What  (18) . . 10 


SECTION  II. 

CONSTRUCTING  EQUATIONS,  OE  FINDING  THEIE  LOCI. 

DEFINITIONS. —A  Continuous  Curve  (19)  ;    Branch  (2O)  ',    Symmetry  (21)  ; 

Independent  and  Dependent  Variables  (25) 10-12 

To  LOCATE  A  POINT  ( 22)  10 

To  CONSTRUCT  AN  EQUATION  (23) 11 

DISCUSSING  AN  EQUATION.  —  What  ;  Intersection  ;  Limits  ;  Symmetry 

(26) 12,  13 

EXAMPLES O..c 11-16 


SECTION  III. 

THE  POINT  IN  A  PLANE. 

DEFINITIONS. — Equations  of  a  Point  (27) 17 

EQUATIONS  or  A  POINT.— What  (28)  ;  In  different  angles,  In  the  axes,  In 

the  origin,  Sch's.  1,  2  ;  Points,  how  designated,  Sch.  3 17 

DISTANCE  BETWEEN  TWO  POINTS. — General  Formulae  (29)  \  Special  cases, 

Cor.  and  Sch 18 

EXAMPLES • 17,  18 


SECTION  IV. 

THE  RIGHT  LINE  IN  A  PLANE. 

DEFINITION. — Equation  of  a  Locus  (3O) 19 

EQUATIONS  OF  A  EIGHT  LINE.— Through  Two  Points   (31)  ;    Through  One 
Point  (32  * ;   Common  Form  (33)  J   Referred  to  Oblique  Axes  (34) ;   Meaning 

of"'""*'",,  Cor.l.  19-21 

x  —x' 

DISCUSSION  ofy  =  ax  -\-  1},  Sch's.  1  and  2 20 

METHODS  OF  CONSTRUCTING  y  =  ax -\- b 21 

Locus  OF  AN  EQUATION  OF  THE  FIRST  DEGREE  (35) 22 

EXAMPLES.  . .  21-23 


SECTION   V. 
OF  PLANE  ANGLES,  AND  THE  INTERSECTION  OF  LINES. 

TANGENT  OF  A  PLANE  ANGLE.— Formulae  for  (36) 23 

EQUATION  OF  A  LINE  MAKING  ANY  GIVEN  ANGLE  WITH  ANOTHER  LINE.  — Com- 


CONTENTS.  IX 

PAGE 

mon  form,  When  passing  through  a  given  point  (37)  I  When  Parallel  to 
a  given  Line,  Common  form,  Passing  through  a  given  point  (38}  ; 
When  Perpendicular  to  the  given  line,  Common  form,  Passing  through  a 

given  point  (39) 24 

EXAMPLES 25,  26 

To  FIND  THE  INTERSECTION  OF  LINES  (4O) 26 

EXAMPLES 26-28 

DISTANCE  FROM  A  POINT  TO  A  LINE  (41)  ;    Between  Parallels,  Cor 28 

EXAMPLES 28,  29 


SECTION  VI. 

OF  THE  CONIC  SECTIONS. 

BOSCOVICH'S  DEFINITION  (42) 29 

To  CONSTRUCT  A  CONIC  SECTION  (43) 29 

DEFINITIONS. — Directrix,  Focus,  Focal  Tangents,  Transverse  Axis,  Conjugate 

Axis,  Latus  Rectum,  Vertices,  Focal  Distances,  Eccentricity  (44) 29,  30 

Axis  of  Hyperbola,  Transverse,  Conjugate,   Conjugate  Hyperbola,  Equi 
lateral  Hyperbola  (47) 32,  33 

EXAMPLES '.  30-34 

BOSCOVICH'S  RATIO  =  ECCENTRICITY  (48) 34 

FUNDAMENTAL  RELATIONS  (45,  46,  49) 30,  31,  34,  35 

To  PASS  A  CONIC  SECTION  THROUGH  THREE  POINTS  (5O) 36 

EXAMPLES 37 

EQUATIONS  OF  CONIC  SECTIONS. — General  Equation  (51)  ;  Referred  to  their 
Axes,  In  terms  of  A  and  e  (52),  Common  Forms  (53,  54,  56,  57,  59)  ', 
Referred  to  Ax's  and  Tangent  at  Vertex  (55,  56,  57)  ',  Of  Conjugate 
Hyperbola  (58) '. 37-40 

COMPARISON   OF   EQUATIONS   OF   ELLIPSE   AND    HYPERBOLA  (60) 41 

Locus  OF  EQUATION  OF  SECOND  DEGREE  (61) 41 

FEATURES  OF  THE  EQUATION  WHICH  CHARACTERIZE  THE  DIFFERENT  CONIC  SEC 
TIONS  (62)  ;  Species  dependent  on  A,  B,  C,  (63)  ;  All  varieties  included 

in  Ay*  -f  C&  +  Dy  -f-  Ex+F=  0  (64) 42,  43 

EXAMPLES 42,  43 

VARIETIES.— Of  Ellipse  (65),  Hyperbola  (66),  Parabola  (67)  ',    Eccentricity 

of  Circle  (68) 43-45 

EXAMPLES 46-49 

EXERCISES  in  producing  various  forms  of  the  equation  of  the  Conic  Sections 

directly  from  the  definition 49-51 

THE  ORIGIN  OF  THE  NAME  CONIC  SECTION  (69) 51 

FIVE  POINTS  IN  THE  CURVE  DETERMINE  A  CONIC  SECTION  (70) 52 

EXAMPLES 52-54 

EXERCISES  in  producing  the  equations  of  Conic  Sections  from  various  defini 
tions  . .  54  -57 


X  CONTENTS. 

SECTION  VII. 

EQUATIONS  OF  HIGHER  PLANE  CURVES. 

PAGE 

DEFINITIONS. — Function  (71)  ;  Classes  of  (72)  ;  Algebraic  (73}  ;  Trigo 
nometrical  (74)  ;  Circular  (75)  ;  Logarithmic  (76*)  ;  Exponential 
(77) 57 

Loci  CLASSIFIED.— Higher  and  Lower,  Algebraic  and  Transcendental  (78, 79)     58 

CISSOID. —Definition  (8O)  ;  Construction  (81)  ;  Origin  of  name,  Sch.  1  ; 
Mechanical  method  of  Constructing,  tick.  2  ;  Equation  of  (82)  ',  Discus 
sion  of  Equation,  Sch.  1  ;  Duplication  of  cube  by  means  of,  Sch.  2  ....  58-60 

CONCHOID. —  Definition  (83)',  Construction  (84)]  Mechanical  Construc 
tion,  Sch.  ;  Equation  of  (85)  ;  Discussion  of  Equation,  Sch.  1  ;  Be 
comes  the  equation  of  circle,  Sch.  2  ;  Trisection  of  an  angle  by  means  of, 
Sch.  3 60-62 

WITCH.— Definition  (86)  ;  Construction  (87)  I  Equation  of  (88)  ;  Dis 
cussion  of  Equation,  Sch 62 

LEMNISCATE.  -  Definition  (80)  i  Construction  (OO)  ;  Equation  of  (01)  ; 
Discussion  of  Equation,  ScJi.  1  ;  How  related  to  Equi-lateral  Hyperbola, 
Sch.  2 63 

CYCLOID.— Definitions,  of  the  Locus,  Generatrix,  Base,  Axis,  (02,  O3)  ;  To 
put  the  Generatrix  in  position  (04)  :  Equations  of  the  Cycloid,  1st  form 
(O5),  2nd  form  (90)  ',  Discussion  of  Equation,  Sch.  to  (05),  and  Cor. 
and  Sch.  1  to  (06) 64,  65 

EQUATIONS  OF  SOME  LOCI  WEITTEN  DIRECTLY  FKOM  THE  DEFINITIONS  (08) ....     66 

NUMBEB  OF  PLANE  CUBVES  INFINITE.— A  few  suggested  (09) 66 


CHAPTER  II. 
THE  METHOD    OF   POLAR    CO-ORDINATES. 

SECTION  I. 

OF  THE  POINT  IN  A  PLANE. 

How  A  POINT  is  DESIGNATED  BY  POLAB  CO-OEDINATES  (10O) 67 

DEFINITIONS. — Pole,  Prime  Radius,  Radius  Vector,  Variable  Angle,  Polar  Co 
ordinates  (1O1) .' 67 

EQUATIONS  OF  A  POINT  (1O2)  ;    Examples 67,  68 

DISTANCE  BETWEEN  TWO  POINTS  (1O3)  ;    Examples 68 


SECTION  II. 

OF  THE  RIGHT  LINE. 

EQUATIONS  OF  THE  RIGHT  LINE. — 1st  form,  2nd  form  (104)  ;    Discussion  of 
1st  form,  Sch.  1  :  Discussion  of  2nd  form,  Sch.  2  ;  Examples 68-70 


CONTENTS.  XI 

SECTION  III. 

OF  THE  CIRCLE. 

PAGE 

EQUATION  when  the  Pole  is  the  Circumference,  and  the  Polar  Axis  is  a  diame 
ter  (1O5)  ;  Discussion,  Sch 70,  71 

GENERAL  POLAB  EQUATION  (1OO)  ;  Discussion,  Sch.  ;  Geometrical  Illus 
tration,  Ex.  5 71-73 

EXAMPLES 72,  73 


SECTION  IV. 

OF  THE  CONIC  SECTIONS. 

POLAB  EQUATION  OF  CONIC  SECTION  (107)  ;  Of  Parabola  (1O8)  ;  Of  El 
lipse  and  Hyperbola  (1O9)  ;  Discussion  of  Equation  of  Parabola,  Sch.  1, 
Of  Ellipse,  Sch.  2,  Of  Hyperbola,  Sch.  3  73-75 

EXAMPLES 75,  76 


SECTION  V. 

OF  HIGHER  PLANE  CURVES. 

POLAB  EQUATION  OF  CISSOLD  (HO)  ;    Discussion,  Sch 76 

POLAB  EQUATION  OF  CONCHOID  (111} ;    Discussion,  Sch 77 

POLAB  EQUATION  OF  LEMNISCATE  (112)  ;    Discussion,  Sch - ...  77 

OF    PLANE    SPIRALS. 

DEFINITIONS.— Of  Spiral,  Measuring  Circle,  Spire  (113) 77 

SPIEAL  OF  AECHIMEDES.— Definition  (114)  ;      Construction   (115)  ;    Equa 
tion  of  (116) 78 

EECIPBOCAL  OB  HYPERBOLIC  SPIBAL  (117)  ',    Equation;    Construction 78 

THE  LITUUS  (118)  ;    Equation  ;     Construction 79 

LOGABITHMIC  SPIBAL  (119)  ',  Definition,  Equation,  Construction  (110) 79 


CHAPTER  III. 
TRANSFORMATION    OF    CO-ORDINATES. 


SECTION  I. 

PASSING  FROM  ONE  SET  OF  RECTILINEAR  AXES  TO  ANOTHER. 

DEFINITIONS. — Transformation  ;  Two  aspects  of  the  Problem  ;  Primitive 
Axes  or  System  ;  New  Axes  or  System  ;  ILLUSTRATIONS  (12O)  ;  Practi 
cal  Advantages,  Sch 80,  81 


Xll  CONTENTS. 

PAGE 

FORMULA  FOR  PASSING  FROM  ONE  RECTILINEAR  SET  OF  AXES  TO  ANOTHKR. — 
General  Formulae  ^122. ;  From  aay  set  to  a  Parallel  set  (123, ;  From 
Rectangular  to  Oblique  (124, ;  From  Rectangular  to  Rectangular  (125); 
From  Oblique  to  Rectangular  (126  >;  The  foregoing  where  the  origin  is 
unchanged  (127);  From  Oblique  to  Rectangular,  when  a  and  a'  sig 
nify  the  angles  which  the  Oblique  or  Primitive  axes  make  with  the  Rect 
angular,  or  New  axis  of  x,  Sch 82-84 

EXAMPLES.  .  .  84-90 


SECTION  II. 

PASSING  FROM  RECTILINEAR    TO  POLAR  CO-ORDINATES,  AND  VICE  VERSA. 

FORMULAE  FOR  PASSING  FROM  RECTILINEAR  TO  POLAR  (128} 90 

FORMULA  FOR  PASSING  FROM  POLAR  TO  RECTILINEAR  (12!)) 91 

EXAMPLES  ..  ....  91,  92 


CHAPTER   IY. 

PROPERTIES    OF   PLANE    LOCI   INVESTIGATED   BY 
MEANS    OF   THE  EQUATIONS   OF    THOSE  LOCI. 


SECTION  I. 

TANGENTS   TO    PLANE  LOCI. 

(a)    BY    RECTILINEAR    CO-ORDINATES. 

DEFINITIONS. — Consecutive  points  (13O);  Tangent  (131};  Tangent  has 
the  same  direction  as  the  Curve,  Cor.  (132) 93 

GEOMETRICAL  SIGNIFICATION  OF  —  (133);  A  Tangent  which  makes  any 
given  angle  with  the  axis  of  x,  Which  is  parallel,  Which  is  perpendicular 
(134);  Signification  of  -jjL  when  the  axes  are  oblique  (135);  EXAMPLES.  93-96 

EQUATIONS  OF  TANGENTS.— General  Equation  (136);  Of  the  Ellipse,  .The 
Hyperbola,  The  Parabola,  The  Circle,  and  other  Examples  ;  The  Intercepts 
of  the  Axes  by  a  tangent  (137),  With  the  axis  of  x  in  Ellipse,  Hyperbola, 
Parabola  ;  Other  Examples  ;  To  draw  a  tangent  to  an  Ellipse  (138),  To 
an  Hyperbola  (139),  To  a  Parabola  (14O) 96-101 

STJBTANGENTS. —Definition  (141)  ;  General  value  of  (142)  ;  Of  an  Ellipse, 
Hyperbola,  Parabola,  other  Examples  ;  Use  in  drawing  tangents 101,  102 

LENGTH  OF  TANGENT. —General  formula  (143);  Of  an  Ellipse,  Hyperbola, 
Parabola 102,  103 

ASYMPTOTES  (rectilinear).— Definition  (144\  Illustrations;  To  examine  a 
curve  for  A, *ymptotest—  General  Method  (145*,  By  Inspection  (148),  By 
Developing  the  function  ( 140  ;  An  Asymptote  the  limiting  position  of  a 
Tangent  (146,  ;  Equation  of  (147)  ',  Examples 103-107 


CONTENTS.  Xlll 

VAGK 

(F>)  TANGENTS  TO  POLAR  CURVES. 

How  DETEEMINED  (ISO) 107 

SUBTANGENT.— Definition  (151)  ;    General  Value  (152) ;    Examples.  ..  107-109 
ASYMPTOTES.— How  Determined  (153) ',    Examples 109,  110 


SECTION  II. 

NORMALS  TO   PLANE   LOCI. 

(a)    BY    RECTANGULAR    CO-ORDINATES. 

DEFINITION  or  NORMAL  (154} 


GENERAL  EQUATION  (155).—  Signification  of  —  ~   (156)  ;    Normal  to  El 

lipse,  Hyperbola,  Parabola,  and  other  Examples   ....................   HO,  HI 

SUBNORMAL.  -Definition  (157)  I    General  Value  (158)  ;    To  Cycloid  (159)  ; 

To  draw  a  Tangent  to  the  Cycloid  (  WO)  ;     To  draw  a  Tangent  making  a 

given  angle  (161)  ;    Examples  ...............  .............  HI,  112 

LENGTH  or  NORMAL  (162)  ;    Examples  .  .  . 

PERPENDICULAR  UPON  A  TANGENT   (163)  ;    From  the  focus  of  a  Parabola 

(164)  ;    Examples  ..............................................   112>  113 


(ft)    NORMALS   TO    POLAR    CURVES. 
SUBNORMAL.— Definition  (165}  ;     General  Value  (166)  \    Examples  ...  113,  114 

LENGTH  OF  NORMAL  TO  POLAR  CUBVE  (167) 

LENGTH  OF  PERPENDICULAR  FROM  THE  POLE  UPON  THE  TANGENT  OF  A  POLAR 
CUKVE  (168) 1U 


SECTION  III. 

DIRECTION   OF  CURVATURE. 

(a)   BY  RECTANGULAR  CO-ORDINATES 
CRITERIA  TOR  DETERMINING  DIRECTION  OF  CURVATURE. 
Sign  of  (169)  ;    Sign  of  (17O) ;    Sign  of  yg  (17D  ; 


(b)   BY  POLAR  CO-ORDINATES. 

DEFINITION  OF  DIRECTION  OF  CURVATURE  OF  POLAR  CURVES  (172) 116 

CRITERIA  FOR  DETERMINING  (173, 174}  ;    Examples H6«  H7 


SECTION  IV. 

SINGULAR  POINTS. 

DEFINITION  AND  ENUMERATION  (175) 

MAXIMA  AND  MINIMA    ORDINATES.— Definition  (176);    To  determine  their 


XIV  CONTENTS. 

PAGE 

position  and  value  (177) ;    A  negative  maximum  or  minimum  (178}  ; 

Examples 118>  119 

POINTS  OF  INFLEXION.— Definition  (179],  Illustration  ;  How  determined  by 
Rectangular  Co-ordinates  (18O)  ;  By  Polar  Co-ordinates  (181)  ;  Ex 
amples  •  119-121 

MULTIPLE    POINTS. —Definition,   Species  (182)  ;     How  determined  (183)  ; 

Examples 121-123 

CUSPS.— Definition,  Kinds  (184)  ;    How  determined  (185)  ;    Examples  124,  125 
CONJUGATE  POINTS. —Definition  (186)  ;     Two  Criteria  (187,  188}  ;    How 

to  examine  a  curve  for  Conjugate  Points  (189)  ;    Examples.  , . 125-127 

SHOOTING  POINTS.— Definition  (190 )  ;    Examples 127,  128 

STOP  POINTS. — Definition  (191) ;    Examples .   128 


SECTION   V. 

TRACING  CURVES. 

DEFINITION  (192) 129 

GENERAL  METHOD  (193,  and  Sch.)  ;     Examples 129-132 

To  TRACE  A  CURVE  OF  THE  SECOND  ORDER.  — By  direct  inspection  of  its  equa 
tion    (194)  ;    Examples ;    By    Transformation  of    Co-ordinates  (195)  ; 

Examples 132-135 

To  TRACE  A  POLAR  CURVE  (196)  ;    Examples 135-137 


SECTION  VI. 

RATE   OF   CURVATURE. 

DEFINITIONS.— Curvature  (197),  Illustration  ;  Oscillatory  Circle  (198),  Il 
lustration  ;  Radius  of  Curvature,  Centre  of  Curvature  (199)  ;  Parameter 
(202) 137-140 

CONTACT. — What,  How  closeness  of  Contact  is  characterized,  Orders  of  Con 
tact  (2OO)t  Geometrical  Illustration  (2O1)  ',  Order  of  Contact  dependent 
upon  Parameters  (203)  ;  Order  of  Contact  of  Right  Line  (2O4>,  Of 
Circle,  Of  Parabola,  Ellipse,  Hyperbola  (2O5,  2O6  >  ;  Restriction  of 
these  statements  (2O7)  J  Contact  of  a  Right  Line  at  Point  of  Inflexion 
(215)  ;  Contact  of  Oscillatory  Circle  at  points  of  Maximum  and  Mini 
mum  Curvature  (216),.  At  the  Vertices  of  the  Conic  Sections  (217)  •  •  139-146 

RADIUS  OF  CURVATURE.  —General  Formula  in  terms  of  Rectangular  Co-ordi 
nates  (2O8)  ;  Signification  of  the  sign  (2O9)  ;  Radius  of  Curvature  of 
the  Conic  Sections,  At  the  vertices  (21O,  212),  Varies  how  (211,  213)  ; 
Centre  of  Curvature  in  the  Normal  (214}  ;  Examples 141-145 

WHEN  OSCULATORT  CURVES  INTERSECT  AND  WHEN  NOT  (218)  ;  When  the  Os- 
oulatory  Circle  Cuts  a  Conic  Section  <219) 146,  147 

RADIUS  OF  CURVATURE  OF  POLAR  CURVES  (22O)  ;  Involving  the  Normal 
(,221)  ;  Examples 147,  148 


CONTENTS.  XV 

PAGE 


SECTION  VII. 

EVOLUTES  AND  INVOLUTES. 

DEFINITION  (222),  Illustration 118 

To  FIND  THE  EVOLUTE  (223)  ',    Examples  ;    The  Evolute  of  a  Cycloid  an 

Equal  Cycloid  (228)  ;    Same  Geometrically,  Fch 149-151 

NORMAL  TO  INVOLUTE  TANGENT  TO  EVOLUTE  (226) 151 

RADIUS    OF    CUEVATUBE    VARIES    AS   ABC    OF    EVOLUTE    (227) 151 

A    CUBVE    DESCBIBED    MECHANICALLY   FBOM    ITS    EVOLUTE    (228) 152 

A   CUBVE    HAS   BUT   ONE   EVOLUTE,  BUT   AN    EVOLUTE   HAS  AN  INFINITE   NUMBEE   OF 

INVOLUTES  (229) 152 


SECTION   Fill. 

ENVELOPES  TO  PLANE  CUEVES. 

DEFINITION  (230),  Illustration 152,  153 

To  FIND  THE  ENVELOPE  (231)  ;    Examples  153-159 

ENVELOPE  TANGENT  TO  THE  INTEESECTING  SEEIES  (232) 154 

CAUSTICS. —General  Equation  (233)  ;  When  the  incident  rays  are  parallel 
to  the  axis  of  the  reflector  (234),  When  perpendicular  (235)  ;  Illustra 
tion  ;  Examples 156-159 


SECTION  IX. 

EECTIFICATION  OF  PLANE  CUEVES. 

DEFINITION  (237) 159 

BY  RECTANGULAR  CO-OEDINATES.— General  Formula  (238)  ;  Examples ; 
Circumferences  of  Circles  are  to  each  other  as  the  radii  (24O)  ;  Value  of 
it  (24:1)  ;  Arc  of  Cycloid  equals  twice  the  corresponding  chord  of  the  gen 
eratrix  (242) 159-162 

BY  POLAR  CO-ORDINATES. — General  Formula  (243) ;    Examples 163,  164 


SECTION  X. 

QUADEATUEE  OF  PLANE  SUEFACES. 

DEFINITION  (244) 164 

BY  RECTANGULAR  CO-OEDINATES.— General  Formula  (245) ;  Examples  ;  Areas 
of  Circles  to  each  other  as  squares  of  radii  •  24(t)  ;  Area  of  Circle  whose 
radius  is  1  (247)  ;  Area  of  Circle  =  \r  X  circumference  (248)  ;  Area  of 
Segment  of  Circle  (249)  ;  Area  of  Ellipse  compared  with  Circumscribed 
and  Inscribed  Circles  (£«50) 164-168 

BY  POLAR  CO-OEDINATES. — General  Formula  (251)  ',    Examples 168,  169 


XVI  CONTENTS. 


SECTION   XL 

QUADRATURE  OF  SURFACES  OF  REVOLUTION. 

DEFINITION  (252)  ;     Illustrations 1G9 

GENERAL  FORMULA  (253)  ',  Examples  ;  Surface  of  a  Sphere  =  4  great  Cir 
cles,  or  Circumference  X  Diameter,  Cor.  1  ;  Area  of  Zone  (254:)  ;  Sphere 
and  Circumscribed  Cylinder,  Sell 1G9-17G 


SECTION  XII. 

CUBATURE  OF  VOLUMES  OF  REVOLUTION. 

GENERAL  FORMULA  (255)  ;  Examples  ;  Volume  of  a  Sphere  =  the  surface 
X  a  radius  (25O)  ;  Volumes  of  Spheres  are  to  each  other  as  the  cubes  of 
their  radii  (257)  5  Volume  of  a  Segment  (258)  ',  Volume  of  Sphere  and 
Circumscribed  Cylinder  (259) 171,  178 


SECTION  XIII. 

EQUATIONS  OF  CURVES   DEDUCED  BY  THE  AID  OF  THE  CALCULUS. 

TRACTRIX.  -Definition  (26O)  ;  '  Equation  (261) 172,  173 

Locus  WHOSE  SUBNORMAL  is  CONSTANT  (262) 173 

Locus  WHOSE  NORMAL  is  CONSTANT  (263) 174 

LOCUS  WHOSE  SUBTANGENT  IS  CONSTANT  (264:) 174 

Locus  WHOSE  SUBNORMAL  VARIES  AS  THE  SQUARE  or  ITS  ABSCISSA  (265) 174 

Locus  WHOSE  AREA  is  TWICE  THE  PRODUCT  OF  ITS  CO-ORDINATES  (266) .  .   174 

Locus  WHOSE  ARC  VARIES  AS  THE  SQUARE  ROOT  OF  THE  THIRD  POWER  oi1  its 
ABSCISSA  (267) '.174 


SECTION  XIV. 

OF  TANGENTS   AND  NORMALS. 

[WITHOUT   THE   AID    OF   THE   CALCULUS.] 

TANGENTS.— General  Method  of  producing  the  equation  of  (268)  ;  Ex 
amples,— Tangent  t  Parabola,  Ex.  I  ;  Ellipse,  Ex.  4  ;  Hyperbola,  Ex.  10  ; 
Tangent  of  the  angle  which  a  tangent  to  a  Conic  Section  makes  with  the 
axis  of  x  (260),  Examples  ;  To  find  the  point  on  a  curve  from  which  a 
tangent  must  be  drawn  to  make  a  given  angle  with  the  axis  of  x,  be  paral 
lel,  be  perpendicular  (27O>  ;  Examples 175-179 

SUBTANGENTS.— Definition  (27 1)  ',  To  find  the  length  of  (272)  ;  Ex 
amples,— Subtangent  in  Parabola,  To  draw  a  tangent  by  means  of  (273)  ; 
Subtangent  of  Ellipse,  To  draw  a  tangent  by  means  of  (274f  275)  ;  Sub- 


CONTENTS.  XVII 

PAGE 

tangent  of  Hyperbola  276 ',  To  draw  a  tangent  by  means  of  277}  ;  Halt 
either  axis  a  mean  proportional  between  its  intercepts  by  a  tangent  and 
ordinate,  Ex.  4  ;  Analogy  between  the  equations  of  the  Conic  Sections  and 
the  equations  of  their  tangents  278) 179-181 

NORMALS.  —  Definition  -.270)  ;  To  produce  the  Equation  of  Normal  (28O)  ', 
Tangent  of  angle  which  Normal  makes  with  axis  of  x  ,281)  ;  Examples, 
Normal  to  Ellipse,  Hyperbola,  Parabola,  Circle  ;  Expressions  for  tangent 
of  the  angle  which  a  Normal  to  a  Conic  Section  makes  with  the  axis  of  x 
<2S2) 181,  182 

SUBNORMALS. — Definition  (283}  ',  Examples  in  the  Conic  Sections,  Is  con 
stant  in  the  Parabola  and  =  p,  To  draw  a  tangent  by  means  of  the  latter 
property,  Etfs  1  and  2 183 

THE  PEBPLDICULAB  FBOM  THE  FOCUS  OF  A  PARABOLA  upon  the  tangent  \284)  ; 
Cor.  (285  ;  To  find  the  focus  of  a  Parabola  when  the  curve  and  its  axis 
are  given,  Also  to  draw  a  tangent  (286,  287,  288) 183 


SECTION  XV. 

SPECIAL  PROPERTIES  OF  THE  CONIC  SECTIONS. 

RADII  VECTORES. — Definition  (280}  ;  Sum  of  in  Ellipse  and  difference  in 
Hyperbola  (290}  ;  Length  of  each  (291}  ;  To  construct  an  Ellipse  and 
Hyperbola  on  this  principle  (292)  ;  Radii  Vectores  make  equal  angles  with 
the  tangent  in  Ellipse  and  Hyperbola  293)  ;  Corresponding  property  in 
Parabola  (296)  ',  Angles  included  by  the  Radii  Vectores  and  Normal,  in 
Ellipse  and  Hyperbola  (294};  To  draw  a  tangent  upon  these  principles,  1st, 
from  a  point  in  the  curve,  2nd,  from  a  point  without  (##5);  Same  problems 
in  reference  to  the  Parabola  (298) 184-187 

THE  RECTANGLE  OF  PERPENDICULARS  FROM  FOCI  UPON  TANGENT  (299} 187 

THE  SEMI-CONJUGATE  AXIS  A  MEAN  PROPORTIONAL  BETWEEN  FOCAL  DISTANCES 
(300) 187 

SUPPLEMENTARY  CHORDS  AND  CONJUGATE  DIAMETERS. — Definition  of  Ordinate 
(30  J),  Of  Supplementary  Chords  (30£),  Of  Conjugate  Diameters  (303)  ; 
Fundamental  property  of  Supplementary  Chords  (3O4,  305)  ;  When 
drawn  on  the  Conjugate  Axis  (306)  ;  When  drawn  from  a  point  in  the 
Conjugate  Hyperbola  (307)  ',  This  property  in  the  Circle  (3O8)  ;  Paral- 

J52 

lelism  of  Sup.   Chords  to  the  axes  (309)  ;    The  —  sign  in  aa  =  —  — 

(3Jf0)  ;  Discussion  of  the  Angle  included  by  Sup.  Chords  (311);  Sup. 
Chords  parallel  to  Tangent  and  Diameter  (312,  313}  ;  To  draw  Tangents 
by  this  property  (314}  ;  Relations  between  Conjugate  Diameters  and  the 

Axes  (318)  ;    Examples 188-194 

OBDINATES.—  Relation  to  each  other,  in  Ellipse  (319},  in  Hyberbola  (322), 
in  Parabola  (333)  ;  Corresponding  properties  of  oblique  ordinates  (325} ; 
Relation  of  an  ordinate  to  the  corresponding  segments  of  its  diameter 
(32O)  ;  Latus  Rectum  a  third  proportional  to  the  axes  (321)  ;  The  rela 
tion  of  ordinates  in  the  circle  (323)  ;  Relation  of  ordinates  to  the  conju 
gate  axis  of  Ellipse  (324)  ;  Parallel  chords  bisected  by  Diameter  (326, 
334)  ;  To  find  the  centre,  axes  and  foci  of  a  Conic  Section  when  the  curv- 


XV111  CONTENTS. 

PAGJt 

ature  is  given  (327,  335)  ;  Ordinates  of  different  Ellipses  on  same  axis 
(328);  Of  Ellipse  aud  Circle  on  same  axis  (329 1  ',  The  Trammel  (33O); 
Ordinates  to  different  ellipses  on  same  Conjugate  Axis  (331)  ;  Of  Ellipse 

and  Inscribed  Circle  (332) 195-199 

ECCENTEIC  ANGLE. —Definition  (336)  :     Sine  and  cosine  of  this  angle  (337); 
Advantages,  Sch. ;     Equation  of   Tangent  to  Ellipse  in  terms  of  this  angle 

(338)  ;      Eccentric     angles  of  the    vertices  of   the    Conjugate    Diameter 

(339)  ;     To    draw  a  Conjugate  Diameter  on  this  principle   (34:0)  ;     Rect 
angle  of  Kadii  Vectors  =  Square  of  Conjugate  Diameter  (341)  ;    Sum  of 
the  Squares  of  Conjugate  Diameters  constant  (342)  ;    Examples 199-201 

THE  INTEKCEPTS  OF  A  SECANT  BETWEEN  THE  HYPEKBOLA  AND  ITS  ASYMPTOTES 
(343)  ;     To  construct  an  Hyperbola  on  this  principle,  Sch 201,  202 


CONTENTS.  XIX 


THE 

INFINITESIMAL    CALCULUS 


INTROD  VCTJON. 

PAGE 

DEFINITIONS.  —  Quantity  (1)  ;  Number  (2)  ;  Discontinuous  and  Continuous 
Number  (3,  4,  5),  Illustiations  ;  An  Infinite  Quantity  (6*)  ;  An  Infini 
tesimal  (7j  5  Caution  (8) .. . 1,  2 

INFINITES  AND  INFINITESIMALS  RECIPROCALS  OF  EACH  OTHEK  (9,  10] 3 

OBDERS  OF  INFINITES  AND  INFINITESIMALS. — What  (11)  ;  Relations  to  each 
other  (12)  3 

AXIOMS  (13,  14,  15,  16,  17,  IS)  ;     Illustrations  ;    Examples 4,  6 

CONSTANTS  AND  VARIABLES. — What  (19,20',',  Any  expression  containing  a  va 
riable  is  a  variable  when  taken  as  a  whole  (21  ',  Distinction  of  Depend 
ent  and  Independent  Variables  \229  23,  24),  Illustration  ;  Equicrescent 
Variable  (25)  ;  Contemporaneous  Increments  (26)  ;  Illustration 6,  7 

FUNCTIONS  AND  THEIR  FORMS. — Definition  of  Function  (27),  Illustration  ; 
Exact  limitation  of  the  term,  Sch.  ;  Functions  classified  as  Algebraic  and 
Transcendental,  and  the  latter  as  Trigonometrical,  Circular,  Logarithmic 
and  Exponential,  with  Definitions  (28,  29,  3O,  31,  32, 33)  ;  Functions 
Explicit  cr  Implicit  (34,  35,  36),  Notation  (37)  ;  Functions  Increasing 
or  Decreasing  (38,  39,  4O)  7-9 

THE  INFINITESIMAL  CALCULUS.— What  (41),  Illustration;  Two  Branches 
(42) 9,10 


CHAPTER   I. 

THE  DIFFERENTIAL    CALCULUS. 


SECTION  I. 

DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS. 

DEFINITIONS.— The  Differential  Calculus  (43)  ;    A  Differential  (44)  ;    Con 
secutive  Values  (45),  Illustrations 11 

NOTATION  for  a  Differential  (46) H 

RULES   FOE   DIFFERENTIATING  ALGEBRAIC  FUNCTIONS. 
EULE  1.— To  Differentiate  a  Single  Variable  (47),  Geometrical  Illustration     12 

EULE  2.— Constant  Factors  (48),  Geometrical  Illustration 12,  13 

EULE  3. —Constant  Terms   (49)  ;     Geometrical  Illustration  ;    An  infinite 

variety  of  functions  may  have  the  same  differential  (5O) 13 

EULE  4.— The  Sum  of  Several  Variables  (51),  Illustration  ;    Character  of 

dx,  dy,  dz,  etc.,  Sch 14 


XX  CONTENTS. 

PAGE 

RULE  5.  —The  Product  of  Two  Variables  (52\  Illustration  ;    Rate  of  Change     14 

RULE  6.— The  Product  of  Several  Variables  (53) 15 

RULE  ?.• — Of  a  Fraction  with  variable  numerator  and  denominator  (54)  , 
With  constant  numerator  \55)  ;    With  constant  denominator, 

Sch 15,  16 

RULE  8. — Of  a  Variable  with  exponent  (56)  ;     Square  Root  (57}  ;     Other 

special  rules,  Sch 16 

EXERCISES  in  differentiating * 16-22 

ILLUSTRATIVE  EXAMPLES  showing  the  significance  of  differentiation 22-25 


SECTION  II. 

DIFFERENTIATION  OF  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS. 

DEFINITION. — Modulus  (58)  25 

To  DIFFERENTIATE  A  LOGARITHM,  Common,  Napierian  (5.9) 25 

To  DIFFERENTIATE  EXPONENTIALS. — With  Constant  base  ,CO),  With  reference 
to  Napierian  logarithms  (61)  ',  When  the  base  of  the  Exponential  is  the 
base  of  the  system  of  logarithms  (02  j  ;  Of  exponential  with  variable  base 

(63) 26 

EXERCISES ' 27,  28 

DIFFERENTIATING  A  VARIABLE  WITH  IMAGINARY  EXPONENT  (65 ) 28 

ILLUSTRATIVE  EXAMPLES  .  28-30 


SECTION  III. 

DIFFERENTIATION  OF  TRIGONOMETRICAL  AND  CIRCULAR  FUNCTIONS. 

OF  TRIGONOMETRICAL  FUNCTIONS. — Of  a  sine  (66)  ;  Of  a  cosine  (07 ),  Signifi 
cance  of  the  sign  (08)  ;  Of  a  tangent  (69)  ;  Of  a  cotangent  (70)  ;  Of 
a  secant  (71)  ',  Of  a  cosecant  (72)  ;  Of  a  versed-sine  (73)  ;  Of  a  co- 

versed-sine  (74) 30-32 

EXERCISES 32 ,  33 

ILLUSTRATIVE  EXAMPLES 34,  35 

OF  CIRCULAR  FUNCTIONS. — In  terms  of  sine  ( 7,5'),  Relation  io  differentiating 
trigonometrical  functions  (70)  ;  In  terms  of  cosine  (77)  ;  In  terms  of 
tangent  (79)  ;  In  terms  of  cotangent  (8O)  ;  In  terms  of  secant  (81)',  In 
terms  of  cosecant  (82)  ',  In  terms  of  versed-sine  (83)  ',  In  terms  of  co- 

versed-sine  (84) 35,  36 

EXERCISES  ;    Geometrical  Illustration 37-39 


SECTION  IV. 

SUCCESSIVE  DIFFERENTIATION  AND  DIFFERENTIAL  COEFFICIENTS. 

SUCCESSIVE  DIFFERENTIATION. — Definitions — Of  successive  differentials  (87), 
Illustrations  ;  Of  Second,  Third,  etc.,  differentials  (88)  ;  To  produce 
successive  differentials  (89) 40,  41 


CONTENTS.  Xxi 

PAGE 

EXEBCISES 41 ,  42 

DIFFERENTIAL  COEFFICIENTS. — A  first,  A  second,  A  third  (90}  ;    Illustration  ; 

Differential  coefficients  generally  variable,  tick 42-44 

EXERCISES ; 43,  44 


SECTION    V. 

FUNCTIONS  OF  SEVERAL   VARIABLES,   PARTIAL  DIFFERENTIATION,  AND 
DIFFERENTIATION  OF  IMPLICIT  AND  COMPOUND  FUNCTIONS. 

FUNCTIONS  or  INDEPENDENT  AND  OF  DKPENDENT  VARIABLES  (91)  ;  Illustra 
tions  ...........................................................  ,  .  44 

DEFINITIONS.—  Partial  Differential  (92}  ;  Total  Differential  (93}  ;  Illustra 
tions  ;  Partial  Differential  Coefficient  (94)  ;  Total  Differential  Coefficient 
(95}  ;  Equicrescence  of  variables,  Sch  .................................  45 

TOTAL  DIFFERENTIAL  EQUALS  THE  SUM  OF  THE  PARTIAL  DIFFERENTIALS  (.97)  ; 
Illustrations  ;  Exercises  ......................  '.  .....................  45-48 

NOTATION  of  Differential  Coefficients  (98)  ................................     48 

TOTAL  DIFFERENTIAL  COEFFICIENT.  —  Of  function  of  two  variables,  Formula, 

(99  )  :     Meaning  of  —  in  such  cases,  and  distinction  between    --     and  —  , 
dx  ]_dx_l         dx 

Sch.  ;     Of  three  variables,  Formula  (1OO)  ;     "When  u  =  f(y,  z,  w),  and 
y  =  (p(x),  z  =  tpi(x),  and  w  =  (p2  (x)  (1O1)  ;     Exercises  ...............  48-50 


IMPLICIT  FUNCTIONS.—  To  differentiate  f(x,  y)  =  0  (102}  ;     Why     -      =  0, 

and  —  ,  or  —  not,  Sch.  ;     Exercises  ......  .   51,  52 

dx        dy 

COMPOUND  FUNCTIONS.  —  Definition,  and  methods  of  expression  (1O3)  ;  To 
differentiate  u=f(y\  when  y  =  tp(x)  (1O4)  ;  Exercises;  To  differen 
tiate  u  =  q>(z},  when  z  =/(x,  y)  (105}  ..............................  52,  53 


SECTION  VL 

SUCCESSIVE  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  INDEPENDENT 
VARIABLES,  AND   OF   IMPLICIT   FUNCTIONS. 

BOTH  VARIABLES  MAY  BE  EQUICRESCENT  (lOft)  ',    Illustration 53 

SUCCESSIVE  PARTIAL  DIFFERENTIALS. — Definition  (J.O7)  5  Notation  (11)8)  ; 
Partial  Differential  Coefficients,  Sch.  ;  Examples  ;  Order  of  differentiation 
unimportant  (1O9)  ',  Examples  ;  To  form  successive  Partial  Differentials 
of  a  function  of  two  Independent  Variables  (HO)  ',  Law  of  the  formula, 
Sdi 54-58 

To  FORM  SUCCESSIVE  DIFFERENTIAL  COEFFICIENTS  OF  AN  IMPLICIT  FUNCTION  OF 
A  SINGLE  VARIABLE  (111}  ;  Examples 58-60 

DERIVED  EQUATIONS. — What;  Orders  of;  First  and  Second  produced  from 
u  =  0  =/,x,  y)  (112)  60,  61 


CONTENTS. 

SECTION  VII. 

CHANGE  OF  INDEPENDENT  VARIABLE. 

PAGE 

WHY  NECESSARY  (113) 61 

FORMS  os  -^,  — - ,  __ ,  when  neither  variable  is  equicrescent  (114)  ;  Ex 
amples  62,  63 

FORMULA  FOR  CHANGING  FROM  x  TO  y  (110) 64 

FORMULA  FOR  INTRODUCING  A  NEW  VARIABLE  0  as  the  equicrescent  (117) 64 

EXAMPLES 65,  66 

To  EXPRESS  THE  PARTIAL  DIFFERENTIAL  COEFFICIENTS  of  u  =f(x,  y),  in  terms 
of  r,  and  0,  when  x  =  cp(r,  0),  and  y  =  <p,  (r,  0)  (118) ;  When  x  =  r  cos  0, 
and  y  =  r  sin  0  (119)  66 


CHAPTER   II. 

APPLICATIONS  OF  THE  DIFFERENTIAL  CALCULUS. 


SECTION  I. 

DEVELOPMENT  OF  FUNCTIONS. 

DEFINITION. — Development,  what  (12O>  ;    Illustration 67 

MACLAURIN'S  FORMULA.— Object  of  (121);  The  Formula  produced  (122)  ; 
How  to  repeat  it  (123)  ;  Other  forms  of  writing  it  (124)  ;  Exercises  in 

application 67-73 

To  DEDUCE  THE  BINOMIAL  FORMULA,  Ex.  2 69 

THE  THEORY  OF  LOGARITHMS.  —  To  Produce  the  Logarithmic  Series, — General, 
Re.  7,  Napierian  (126)  ;  Adapted  to  computation  (127;',  Relation  of 
logarithms  of  the  same  number  in  different  systems  (128)  ;  To  find  the 
Modulus  of  the  Common  System  (129,  ;  To  compute  a  table  of  common 
logarithms  (13O);  To  find  the  relation  of  the  modulus  to  the  base,  Ex.  8  ; 
To  find  the  N.ipierian  base  (131}  ;  To  produce  the  exponential  series, 
Ex.  9  ;  To  obtain  the  Napierian  bass  from  the  exponential  series  (132) . .  70-72 

To  FIND  THE  YALUE  OF  it  from  y  =  tan-'  a:,  Ex.  10,  and  tick 73 

MACLAURIN'S  FORMULA  NOT  APPLICABLE  to  all  forms  of  functions  (133)  ;  Ex 
amples  ;  Occasion  of  the  inapplicability  (134) 73,  74 


TAYLOR'S  FORMULA. -Object  of  (135)  ;    Lemma,  If  u  ==  f(x  +  y\  ^  = 

(136)  ;  Coefficients  depend  upon  the  form  of  the  function,  Sch.  ;  Ex 
amples  ;  Formula  Produced  (137)  ;  How  to  state  it  (138)  ;  Exercises 
in  applying  it ;  Use  in  developing  y  =f(x),  when  x  takes  an  increment 

(139) .' ,.    ...   74-78 

TAYLOR'S  FORMULA  SOMETIMES  FAILS  for  certain  values  of  the  variable  (14O) 
Examples  ;  Distinction  between  the  failure  of  Maclaurin's  and  of  Taylor's 
formulae,  Sch.  2 .  78,  79 


CONTENTS.  XXili 


SECTION  II. 

EVALUATION  OF  INDETERMINATE  FORMS. 

PAOH 

THE  INDETEEMINATE  FORMS  Enumerated  (141)  ;  Illustration  ;  All  reducible 
to  one  ;  From  what  the  apparent  indetermination  sometimes  atises  (142); 
Illustration 80 

EVALUATION  OP  ^  (143) ;    Examples 81,  82 

EVALUATION  or  —  (144)  ;     Examples 83,  84 

00 

EVALUATION  OF  0  x  <x>  (145) ;    Examples 84,  85 

EVALUATION  OP  oo  —  oo  (146)  ;    Examples 85,  86 

EVALUATION  OP  0°,  <x°,  and  1 x  (147)  ',    Examples 86,  87 


SECTION  III. 

MAXIMA  AND  MINIMA   OF  FUNCTIONS  OF  ONE  VARIABLE. 

DEFINITIONS. — Maximum  (148}  ;  Illustrations  ;  Minimum  (149)  ;  Illus 
trations  ;  Same  function  may  have  several  maxima,  and  several  minima 
values  (15O)  ;  Geometrical  Illustrations  ;  Technical  use  of  the  terms 
(151) 88,  89 

AT  A  MAXIMUM  OB  MINIMUM  -j-  =  0,  or  oo  (152)  ;    Geometrical  Illustrations  ; 

Not  all  values  arising  from  -J?  =  0  or  oo,  correspond  to  maxima  or  min 
ima 89-91 

SIGN  OP  -p2  AT  POINTS  OP  MAXIMA  AND  MINIMA  (153) 91 

ORDINARY  METHOD  or  PROCEDURE  (154)  ;  Axioms  which  facilitate  the  work 
(155)  ;  Examples  ;  Failure  of  the  ordinary  method,  Ex.  10  ;  Method 
in  such  cases,  Ex.  10,  and  (156)  ;  Geometrical  Problems 91-101 


CHAPTER   III. 

THE   INTEGRAL    CALCULUS. 


SECTION  I. 

DEFINITIONS  AND  ELEMENTARY  FORMS. 

DEFINITIONS.—  The  Integral  Calculus  (157)  ;  An  Integral  (158) ;  Integra 
tion  (159)  ;  Sign  of  Integration  (16O)  ;  Illustrations  of  preceding  defi 
nitions '. 102 

No  SUCH  THING  AS  A  PROCESS  OP  INTEGRATION  (161) 103 

THREE  ELEMENTABT  PROPOSITIONS.— Constant  factors  (162)  ;    Algebraic  sum 


XXIV  CONTENTS. 

PAGE 

of  differentials  ( 163)  ;    The  Indeterminate  Constant  (164)  ;    Method  of 

disposing  of  this  constant,  Sch 102,  103 

Two  ELEMENTARY  RULES. — For  a  monomial  differential,  or  a  differential 
which  may  be  so  treated  (165)  ;  Examples  ;  For  differential  of  a  loga 
rithm  (166) 104 

ELEMENTARY  FORMS  (167) 105 

SUBORDINATE  ELEMENTARY  FORMS  (168)  ',    How  obtained 106,  107 

LOGARITHMIC  TRIGONOMETRICAL  FORMS  (169) 108 

EXAMPLES  in  application  of  Elementary  Forms • 108-117 

SIMPLE  EXPEDIENTS  for  reducing  to  elementary  forms. — Introduction  of  a 
constant  factor  (17  O),  and  Ex's  following  ;  Caution,  Sch.  ;  Reduction  of 
a  differential  by  transfer  of  variable  factor  (171)  >  How  the  constant  is 
written  in  logarithmic  integrals,  Sch 108-117 


SECTION  II. 

EATIONAL  FRACTIONS. 

DEFINITION. — Rational  Fraction  (172) 117 

NUMERATOR  MAY  BE  MADE  OF  LOWFR  DIMENSIONS  THAN  THE  DENOMINATOR  (173)  118 
SEPARATION  INTO  PARTS  by  Indeterminate  Coefficients. — When  the  Denomina 
tor  is  resolvable  into  Real  and  Unequal  factors  (174)  ;  When  resolvable 
into  Real  and  Equal  factors  (175)  ;  When  the  factors  are  Imaginary,  Sch. ; 
When  the  factors  are  Real,  Equal  Quadratics  (176)  ;  How  to  find  the  fac 
tors  (177)  ;  Forms  on  which  the  integration  depends  in  these  cases 
(178)  ;  Examples 118-123 


SECTION  III. 

RATIONANIZATION. 

BINOMIAL  DIFFERENTIALS.—  Reducible  to  the  form  xm(a  -\-  bX")T>dx  (18O)  ; 
Can  be  rationalized,  1st,  (181}  ;  2nd,  (182)  ;  Not  always  expedient 
(183)  ;  When  either  m  or  p  is  a  positive  integer  (184)  ;  Examples. .  124-127 

IRRATIONAL  FRACTIONS. — With  none  but  monomial  surds  (185)  ;    No  surd 

m 

except  of  the  form  ta  -}-  &#)"  (186)  ;    How  rationalized  when  the  surd  is 
1     of  the  form  v/a  -j-  bx  ±  &  (187)  ;    Examples 127-129 


SECTION  IV. 

INTEGRATION  BY  PARTS. 

FORMULA  FOR  INTEGRATION  BY  PARTS  (188) 130 

FORMULA  OF  REDUCTION.— Formula  ^  (189) ;  Formula  ££  (19O)  ;  Form 
ula  <JJ  (191)  ;  Formula  ?£>  (192)  ;  Sometimes  fail,  Sch.  ;  Ex 
amples 130-134 


CONTENTS.  XXV 

PAGE 

LOGARITHMIC  DIFFERENTIALS. — Of  the  form  dy  =  X  •  log*xdx  (103)  ;  Ex 
amples  134,  135 

EXPONENTIAL  DIFFERENTIALS. — Of  the  form  dy  =  Wa^dx  (194) ;  Ex 
amples  135,  136 

SPECIAL  FORMS  OF  EXPONENTIALS  136,  137 

TRIGONOMETRICAL  DIFFERENTIALS. —Of  the  forms  dy  =  siumx  dx,  dy  =  cos«x  dx, 
dy  =  sinm  x  cos"  xd.v  (195)  ;  When  this  process  is  applicable  and  the  final 
forms,  (196)  ;  Special  method  when  the  exponents  are  even  (page  140)  ; 

Of  the  form  dy  =  B-^-£-d#,  Ex.  17  ;    Examples 137-140^ 

Of  the  forms  dy  =  xn  smxdx  and  dy  =  xn  cosxdx  (197)  '•>    Examples .    ...  141 
Of  the  form  dy  =  sinw  x  cos"  x  dx  integrated  in  terms  of  multiple  arcs  (198)', 
Examples    141,  142 

CIRCULAR  DIFFERENTIALS. — Of  the  forms  dy  =f(x)  sin-1  xdx,  f(x)  cos-*1  x  dx, 
etc.  (199) :  Examples 142,  143 

OF  THE  FORMS  dy  =  e«*  sin*xdx,  dy  =  e«*  cos"  cede  (20O)  ;    Examples  . .  143,  144 

OF  THE  FORM  dy  = ~ (2O1)  .  144,  145 

(a  -f-  6  cos  x)n  ^ 


SECTION  V. 

INTEGRATION   BY   INFINITE   SERIES. 

OCCASION  FOR  THIS  METHOD  (2O2) 146 

EXAMPLES  illustrating  the  method 146,  147 


SECTION   VI. 

SUCCESSIVE  INTEGRATION. 

GENERAL  PROBLEM  (203)  ;     Examples ;     The   constants    strictly  general, 

Sch.  1  ;    Condition  of  complete  integration,  Sch.  2 147,  148 

THE  nth  INTEGRAL  DEVELOPED  by  Maclaurin's  Formula  (2O4)  ;    Example  . . .  149 


SECTION  VII. 

DEFINITE  INTEGRATION   AND  THE  CONSTANTS   OF  INTEGRATION. 

DEFINITIONS.— An  Indefinite  Integral  (2O&)  ;  A  Corrected  Integral  (2O6) ; 
Integration  between  Limits  (2O7)  ',  A  Definite  Integral  (2O8)  ;  Ex 
amples 150,  151 

DISPOSING  OF  THE  CONSTANT  OF  INTEGRATION. — Two  methods  (2O9) ;  Ex 
amples  151,  152 


INTRODUCTION. 


A  BRIEF  SURVEY  OF  THE  OBJECT  OF  PURE  MATHEMATICS 
AND  OF  THE  SEVERAL  BRANCHES. 

1.  Pure  Mathematics  is  a  general  term  applied  to  several 
branches  of  science,  which  have    for   their  object  the  investigation 
of  the   properties   and  relations   of  quantity — comprehending  num 
ber,  and  magnitude  as  the  result  of  extension — and  of  form. 

2.  The  Several  Branches  of  Pure  Mathematics  are  Arith 
metic,  Algebra,  Calculus,  and  Geometry. 

3.  Arithmetic,    Algebra,   and    Calculus   treat    of    number ;    and 
Geometry  treats  of  form  and  magnitude  as  the  result  of  extension. 

4.  Quantity  is  the  amount  or  extent  of  that  which  may  be 
measured  ;  it  comprehends  number  and  magnitude. 

The  term  quantity  is  also  conventionally  applied  to  symbols  used 
to  represent  quantity.  Thus  25,  m,  xi,  etc.,  are  called  quantities, 
although,  strictly  speaking,  they  are  only  representatives  of  quan 
tities. 

Sen.  1. — It  is  not  easy  to  give  a  philosophical  account  of  the  idea  01;  ideas 
represented  by  the  word  Quantity  as  used  in  Mathematics  ;  and,  doubtless, 
different  persons  use  the  word  in  somewhat  different  senses.  It  is  obviously 
incorrect  to  say  that  "Quantity  is  anything  which  can  be  measured." 
Quantity  may  be  affirmed  of  any  such  concept ;  nevertheless,  it  is  not  the 
thing  itself,  but  rather  the  amount  or  extent  of  it.  Thus,  a' load  of  wood,  or 
a  piece  of  ground,  can  be  measured  ;  but  no  one  would  think  of  the  wood 
or  the  ground  as  being  the  quantity.  The  quantity  (of  wood  or  ground)  is 
rather,  the  amount  or  extent  of  it.  The  word  is  very  convenient  as  a  general 
term  for  mathematical  concepts,  when  we  wish  to  speak  of  them  without 
indicating  whether  it  is  number  or  magnitude  that  is  meant.  Thus  we  say, 
"  in  represents  a  certain  quantity,"  and  do  not  care  to  be  more  specific. 

As  applied  to  number,  perhaps  the  term  conveys  the  idea  of  the  whole, 
rather  than  of  that  whole  as  made  up  of  parts.  It  is,  therefore,  scarcely 
proper  to  speak  of  multiplying  by  a  quantity ;  we  should  say,  by  a  number. 


2  INTHODUCTION. 

On  the  other  hand,;  whsn  w>3.  apply,  the  term  quantity  to  magnitude,  it  is  with 
the  idea  that  magnitude  may  be  measured,  and  thus  expressed  in  number. 

The  distinction  between  quantity  and  number  is  marked  by  the  questions, 
"  How  much  ?"  and  "How  many  ?" 

SCH.  2.  —  So,  also,  the  word  Magnitude,  as  used  in  mathematics,  is  not 
easily  defined.  Sometimes  it  has  reference  to  quantity  in  the  aggregate,  or 
mass,  and  sometimes  to  the  relation  which  one  quantity  bears  to  another. 
Thus,  we  speak  of  a  line,  a  surface,  or  a  solid,  as  a  magnitude,  simply  mean 
ing  thereby  that  these  have  extent,  —  are  extended.  A  circle,  a  triangle,  a 
cube,  are  magnitudes,  —  L  e.,  they  have  extension.  Again,  we  speak  of  the 
magnitude  of  a  circle,  meaning  its  size,  —  area  as  compared  with  some  other 
surface.  The  magnitude  of  a  line  is  expressed  by  telling  how  many  times 
it  contains  another  line  of  known  length.  In  like  manner  the  magnitude 
of  a  surface  or  a  volume  is  made  known  by  comparing  the  surface  with  some 
unit  of  surface,  and  the  volume  with  some  unit  of  volume.  In  one  aspect, 
therefore,  number  is  an  expression  for  the  ratio  of  magnitudes. 


is  quantity   conceived  as  made   up   of  parts,  and 
answers  to  the  question,  "How  many?" 

ILLUSTRATION.  —  Thus,  a  distance  is  a  quantity  ;  but  if  we  call  that  distance  5,  we 
convert  the  notion  into  number,  by  indicating  that  the  distance  under  consid 
eration  is  made  up  of  parts.  Now,  the  distance  may  be  just  the  same,  whether 
we  consider  it  as  a  whole,  or  think  of  it  as  5,  —  L  e.,  as  made  up  of  5  equal  parts. 
Again,  ra  may  mean  a  value,  as  of  a  farm.  We  may  or  may  not  conceive  it  as  a 
number  (as  of  dollars).  If  we  think  of  it  simply  in  the  aggregate,  as  the  worth 
of  a  farm,  m  represents  quantity  ;  but  if  'we  think  of  it  as  made  up  of  parts  fas 
of  dollars),  it  is  a  number. 

6.  Number  is  of  two  kinds,   Discontinuous  and  Contin 
uous. 

7.  Discontinuous  Number  is  number  conceived  as  made  up 
of  finite  parts  ;    or  it  is  number  which  passes  from  one  state  of  value 
to  another  by  the  successive  additions  or  subtractions  of  finite  units, 
—  i.  e.,  units  of  appreciable  magnitude. 

8.  Continuous  Number  is  number  which  is  conceived  as 
composed  of  infinitesimal  parts  ;  or  it  is  number  which  passes  from 
one  state   of  value  to  another  by  passing  through  all  intermediate 
values,  or  states. 

ILL.—  The  method  of  conceiving  number  with  which  the  pupil  has  become 
familiar  in  arithmetic  and  algebra,  characterizes  discontinuous  number.  Thus 
the  number  13  is  conceived  as  produced  from  5  by  the  successive  additions  of 
finite  units,  either  integral  or  fractional.  In  either  case  we  advance  by  succes 
sive  steps  of  finite  length.  If  we  Bay  5,  G,  7,  etc.,  till  we  reach  13,  we  pass  by 
one  kind  of  steps  ;  and,  if  we  say  5.1,  5.2,  5.3,  etc.,  till  we  reach  13,  we  pass 


OBJECT   OF   PURE   MATHEMATICS. 

by  another  sort  of  steps  (tenths),  but  as  really  by  finite 

ones.     If,  however,  we  call  the  line  AB,  Fiy.  1,  x,  ./ 

and  CD,  x',  and  conceive  AB  to  slide  to  the  po 
sition  CD,  increasing  in  length  as  it  moves  so  as  to 
keep  its  extremities  in  the  lines  OM  and  ON, 
it  will  pass  by  infinitesimal  elements  of  growth  from 
the  value  x,  to  the  value  x';  or,  it  will  pass  from 

one  value  to  the  other  by  passing  through  all  inter-  FIG.  1. 

mediate  values,  and  thus  becomes  an  illustration  of  continuous  number. 
Again,  if  the  line  AB,  Fig.  2,  be  considered  as 


generated  by  a  point  moving  from  A  to  B,  and  we  A.          C  B 

call  the  portion  generated  when  the  point  has  reached  FIG.  2. 

C,  z,  and  the  whole  line  x',  x  will  pass  to  x'  by  re 
ceiving  infinitesimal  increments,   or  by  passing  through  all  states  of  value  be 
tween  x  and  x'. 

A  surface  may  be  considered  as  generated  by  the  motion  of  a  line,  and  thus 
afford  another  illustration  of  continuous  number. 
Thus  let  the  parallelogram  AF  be  conceived  as 
generated  by  the  right  line  A  B  moving  parallel  to 
itself  from  A  B  to  E  F.  When  A  B  has  reached 
the  position  CD,  call  the  surface  traced,  namely 
A  BCD,  x,  and  the  entire  surface  ABEF,  x'; 

then    will   x   pass  to    x'   by   receiving  infinitesimal  increments,  or  by  passing 
through  all  intermediate  values. 

Finally,  as  volumes  may  be  conceived  as  generated  by  the  motion  of  planes, 
all  geometrical  magnitudes  afford  illustrations  of  continuous  number. 

We  usually  conceive  of  time  as  discontinuous  number,  as  when  we  think  of 
it  as  made  up  of  hours,  days,  weeks,  etc.  But  it  is  easy  to  see  that  suoh  is  not 
the  way  in  -which  time-  actually  grows.  A  period  of  one  day  does  not  grow  to 
be  a  period  of  one  week  by  taking  on  a  whole  day  at  a  time,  or  a  whole  hour, 
or  even  "a  whole  second.  It  grows  by  imperceptible  increments  (additions). 
These  inconceivably  small  parts  of  which  continuous  number  is  made  up  are 
called  Infinitesimals. 

Motion  and  force  afford  other  illustrations  of  continuous  number.  In  fact, 
the  conception  which  regards  number  as  continuous,  will  be  seen  to  be  less 
artificial — more  true  to  nature — than  the  conception  of  it  as  discontinuous. 

0.  Arithmetic  treats  of  Discontinuous  Number,  of  its  nature 
and  properties,  of  the  various  methods  of  combining  and  resolving 
it,  and  of  its  application  to  practical  affairs. 

For  an  outline  of  the  topics  of  Arithmetic,  see  the  Complete  School  Algebra, 
ABT.  9. 

10.  Algebra  treats  of  the  Equation,  and  is  chiefly  occupied 
in  explaining  its  nature,  and  the  methods  of  transforming  and 
reducing  it,  and  in  exhibiting  the  manner  of  using  it  as  an  instru 
ment  for  mathematical  investigation. 


4  INTRODUCTION. 

For  a  full  account  of  the  province  of  Algebra,  see  the  Complete  School  Algebra, 
ART.  10. 

11.  Calculus   (The  Infinitesimal  Calculus)   treats  of    Continu 
ous  Number,  and  is  chiefly  occupied  in  deducing    the    relations  of 
the  infinitesimal  elements  of  such  number  from  given  relations  be 
tween  finite  values,  and  the  converse  process,   and  also  in  pointing 
out   the  nature  of   such    infinitesimals    and  the  methods   of   using 
them  in  mathematical  investigation. 

12.  Geometry  treats  of  magnitude  and  form  as  the  result   of 
extension  and  position. 

Sen.  1. — The  principal  divisions  of  the  science  of  Geometry  are  : 

1.  The   Ancient,    Platonic,  Special,   Graphic,  or    Direct  Geometry   (the 
common  Geometry  of  our  schools),   including  Trigonometry,   Conic  Sec 
tions,  and  all  other  geometrical  inquiries  conducted  upon  these  methods. 

2.  The  Analytical,    Modern,  Cartesian,  General,  or  Indirect   Geometry 
(the  theme  of  this  volume),  and 

3.  Descriptive  Geometry. 

SCH.  2. — The  first  system  of  geometrical  investigation  probably  took  its 
rise  as  a  science  in  the  school  of  Plato  (about  400  B.  C.),  and  was  brought 
almost  to  its  present  state  of  perfection,  as  far  as  its  methods  are  concerned, 
by  the  time  of  Euclid  (about  300  B.  C.);  hence  it  is  called  the  Ancient  or 
Platonic  Geometry.  As  the  argument  is  earned  forward  by  a  direct  inspec 
tion  of  the  forms  (figures)  themselves,  delineated  before  the  eye,  or  held  in 
the  imagination,  it  is  called  the  Direct  or  Graphic  method.  Inasmuch 
as  it  discusses  particular  instead  of  general  problems  it  is  properly  charac 
terized  as  Special.  With  this  method  of  geometry  the  student  is  supposed 
to  be  acquainted  before  commencing  the  study  of  this  volume. 

The  fundamental  notion  of  the  Modern  Geometry  (a  system  of  co 
ordinates),  was  developed  by  Des  Cartes  in  the  earlier  part  of  the  17th 
century,  and  hence  the  names  Modern  or  Cartesian.  The  term  Analytical 
Las  come  to  be  applied  in  mathematics  in  the  sense  of  Algebraical,  all 
investigations  carried  forward  chiefly  by  the  aid  of  Algebra  being  called 
Analytical.  This  use  of  the  term  is  quite  unfortunate,  inasmuch  as  the 
processes  of  Algebra  are  no  more  analytical,  in  the  true  sense  of  that 
term,  than  are  those  of  the  Special  Geometry.  Again,  as  a  name  for  the 
General  Geometry,  even  if  used  in  the  sense  of  algebraic,  the  term  docs 
not  distinguish  the  system  from  any  other  application  of  algebra  to 
geometry. 

The  true  character  of  the  Modern  Geometry  is  expressed  by  the  terms 
Indirect,  and  General.  This  system  of  geometrical  reasoning  proposes 
the  solution  of  general  problems,  and  effects  its  purpose  by  first  translat 
ing  geometrical  forms  into  equations,  then  carrying  forward  the  investi 
gation  by  means  of  these  equations,  and  finally  returning  to  the  geomet 
rical  forms  by  a  re-translation.  The  indirectness  of  this  method  is  appa- 


OBJECT   OF  PUKE  MATHEMATICS.  5 

rent,  and  might  seem,  in  itself,  a  serious  objection  ;  but  it  is  found  to 
be  of  great  advantage,  inasmuch  as  it  makes  the  discussions  much  more 
comprehensive  (general}.  To  illustrate  this  general  (comprehensive)  char 
acter  of  its  discussions,  we  have  only  to  notice  some  of  its  problems.  Thus 
the  Special  Geometry  discusses  the  problem  of  the  tangent  to  a  circle, 
and,  on  an  independent  basis,  investigates  the  properties  of  a  tangent  to 
any  other  curve,  making  a  special  problem  with  respect  to  each  separate 
curve  studied.  On  the  other  hand,  the  General  Geometry  proposes  the 
problem  in  this  way  :  To  find  a  formula  sufficiently  general  to  embrace  the 
properties  of  tangents  to  ALL  plane  curves  ; — in  technical  language,  To  find 
the  equation  of  the  tangent  to  any  plane  curve.  Again,  in  the  Special  Ge 
ometry,  the  area  of  a  circle  is  obtained  (approximately).  But  the  General 
Geometry  proposes  to  investigate  the  problem  on  a  broader  basis,  and  find 
a  formula  which  shall  be  applicable  in  finding  the  area  of  any  plane  curve. 

13.  Descriptive  Geometry  is  that  system  of  geometry 
which  seeks  the  graphic  solution  of  geometrical  problems  by  means 
of  projections  upon  auxiliary  planes. 

This  is  the  ordinary  definition  of  the  Descriptive  Geometry,  and  it  would  be 
out  of  place  to  attempt  any  elucidation  of  it  here. 

SCH. — From  the  definition  of  Geometry,  as  well  as  from  the  detailed 
study  of  its  propositions,  it  will  be  seen  to  embrace  two  classes  of  prob 
lems;  viz.,  Problems  relating  to  Position,  and  Problems  relating  to  Magni 
tude.  Problems  of  the  latter  class  were  solved  by  the  aid  of  algebra 
before  the  time  of  Des  Cartes  ;  but  it  was  reserved  for  him  to  invent  a 
method  by  which  problems  of  both  kinds  could  be  so  discussed.  This 
system  constitutes  the  foundation  of  the  General  Geometry. 

14:.  The  inquiries  in  the  General  Geometry  may  be  divided  into 
two  classes,  viz. : 

1.  Concerning  Plane  Loci, 

2.  Concerning  Loci  in  Space. 

In  accordance  with  this  division  the  present  treatise  is  divided 
into  Two  Books* 

SCH. — This  division  is  found  especially  convenient  when  the  subject 
is  treated  by  the  aid  of  the  Calculus,  as  it  corresponds  to  the  distinction 
between  functions  of  a  single  variable,  and  functions  of  two  variables. 

*  The  Second  Book  is  reserved  for  another  volume,  which  will  also  contain  au  advanced  course 
in  the  Calculus. 


BOOK  I. 

OF    PLANE    LOCI 


CHAPTER  1. 

THE  CARTESIAN  METHOD  OF   CO-ORDINATES. 


SECTION  L 

Definitions  and  Fundamental  Notions, 

1.  The  term  Locus  as  used  in  geometry  is  nearly  synonymous 
with  geometrical  figure,  yet  having  a  latitude  in   its  use  which  the 
latter  term  does  not  possess.     The    locus    of    a    point    is  the  line 
(geometrical  figure)  generated  by  the  motion  of  the  point  accord 
ing  to  some  given  law.     In  the  same  manner,  a  surface  is  conceived 
as  the  locus  of  a  line  moving  in  some  determinate  manner. 

2.  The  General  Geometry  is  a  system  of  geometrical  in 
vestigation  in  which  the  loci  under  consideration  are  represented  by 
equations,    and  the  inquiries    carried    forward  by    means  of    these 
equations,  the  final  object  being  the  discussion  of  general  problems. 

[NOTE. — While  it  is  true  that  the  only  way  to  obtain  a  full  comprehension  of  the  nature  of  a 
science  is  by  the  detailed  study ,of  its  parts,  it  is,  nevertheless,  important,  at  the  outset,  to  com 
prehend  :is  clearly  as  possible  the  general  aim  of  the  science,  in  order  that  the  tendency  of  the 
several  steps  in  our  progress  may  be  perceived,  and  the  symmetry  and  unity  of  the  whole  may 
appear.  According  to  our  definition  it  will  be  our  first  purpose  to  exhibit  a  scheme  by  which 
points,  lines  straight  and  curved,  the  magnitude  of  Bangles,  surfaces,  etc.,  which  we  have  char 
acterized  as  "  geometrical  forms  "  (loci),  may  be  represented  by  equations.  This  will  be  done  in 
Section  1st  of  this  chapter.  Section  2nd  will  then  exhibit  a  method  of  constructing  the  geometrical 
figure  represented  by  any  given  equation.  Then  will  follow  a  series  of  sections  showing  how  the 
equations  of  loci  are  derived  from  the  definitions  of  the  figures.  This  series  of  sections  comprises 
what  may  be  termed  the  translation  of  geometrical  forms  into  algebraic  equations,  and  will  answer 
such  questions  as:  "What  equations  represent  points?  "What  straight  lines?  What  circles? 
What  ellipses?  etc.,  etc."  Section  2nd,  which  shows  how  equations  are  translated  into  geometrical 
forms,  might,  perhaps,  with  strict  logical  propriety,  follow  instead  of  precede  this  series  of  sections  ; 
but  it  is  thought  the  present  arrangement  will  promote  clearness  of  conception.  The  first  three 
chapters  will  be  seen  to  be  preparatory.  It  is  not  their  purpose  to  develop  geometrical  truths,  but 


DEFINITIONS  AND  FUNDAMENTAL  NOTIONS.  7 

simply  to  prepare  instruments  (the  equations  of  loci)  to  be  subsequently  used  in  conducting  geo. 
metrical  inquiries.  In  the  fourth  chapter  it  will  be  our  purpose  to  show  how  geometrical  truth 
can  be  developed  by  means  of  these  equations.  ] 

3.  A  device  by  means  of  which  we  are  enabled  to  represent  loci 
by  equations  is  called  a 

METHOD  OF  CO-ORDINATES. 

4.  There  are  two  systems  of  co-ordinates  in  common  use,  viz.: 

1.  The  system  of  Rectilinear  Co-ordinates, 

2.  The  system  of  Polar  Co-ordinates. 

5.  There  are  two  varieties  of  the  rectilinear  system  of  co-ordi 
nates,  the  rectangular  and  the  oblique.     (In  our  study,  the  rectan 
gular  system  will  always  be  used  unless  otherwise  specified. ) 

0.  In  order  to  locate  a  point  in  a  plane  by  the  method  of  recti 
linear  co-ordinates,  two  lines  intersecting  each  other  are  assumed 
as  fixed  in  position.  These  lines  are  called  Axes  of  Reference, 
or,  simply,  The  Axes.  The  system  i&  called  rectangular  or 
oblique,  according  as  these  lines  make  a  right  or  an  oblique  angle 
with  each  other. 

7.  One  of  these  axes  is  called  the    Axis  of  Abscissas,  and 
the  other  is  called  the  Axis  of  Ordinates. 

8.  The  Origin  is  the  intersection  of  the  axes. 

9.  The  Co-ordinates  of  a  point  are  its  distances  from  the 
axes,  the  distance  to  either  axis  being  measured  on  a  line  parallel 
to  the  other,  or  on  that  other  axis. 

10.  The  Abscissa   of  a   point    is  the   co-ordinate   which   is 
measured  parallel  to  or  on  the  axis  of  abscissas,  and  is  the  distance 
of  the  point  from  the  axis  of  ordinates  measured  on  a  line  parallel 
to  the  axis  of  abscissas. 

11.  Tfie  Ordinate  of  a  point  is  the   co-ordinate   which    is 
measured  parallel  to  or  on  the  axis  of  ordinates,  and  is  the  distance 
of  the  point  from  the  axis  of  abscissas  measured  on  a  line  parallel 
to  the  axis  of  ordinates. 

Sen.  1.— These  lines,  when  spoken  of  separately,  should  be  distinguished 
as  abscissa  and  ordinate ;  but,  when  taken  together,  they  are  called  co 
ordinates. 


8  THE  CARTESIAN  METHOD  OF  CO-ORDINATES. 

ILI* — Definitions  3  to  11  may  be  illus 
trated  thus  :  Let  the  plane  in  which  the  loci 
are  situated  be  represented  by  the  surface 
of  the  paper,  Fig.  4.  In  this  plane  assume 
two  fixed,  indefinitely  extended,  straight 
lines,  as  XX'  and  Y  Y',  intersecting  each 
other  at  A,  and  to  which  all  points  in  the 
plane  are  to  be  referred.  These  lines  are 
the  Axes,  and  A  is  the  origin,  i.e.,  the  point 
at  which  the  co-ordinates  are  conceived  to 
originate,  and  from  which  they  are  reck 
oned.  One  of  these  lines,  as  XX' (in  ordinary  use  the  horizontal  one)  is  called 
the  Axis  of  Abscissas,  because  abscissas  are  reckoned  oa  it  ;  and  the  other, 
YY',  is  for  a  like  reason  called  the  Axis  of  Ordinates.  Ths  system  is  called 
rectangular  or  oblique  according  as  YAX  is  a  right  or  an  oblique  angle.  It  is 
evident  that  we  can  now  define  the  position  of  any  point  in  this  plane  by  giving 
its  distances  from  these  two  fixed  lines,  or  axes.  For  convenience,  we  measure 
these  distances  on  lines  parallel  to  the  axes.  (In  the  case  of  restangular  axes, 
the  co-ordinates  will  become  the  perpendicular  distances  of  points  from  the 
axes.)  Thus  the  location  of  the  point  P  is  determined  by  giving  the  lengths 
of  PE,  the  abscissa  of  P,  and  of  PD,  the  ordinate  of  the  point.  Usually, 
A D  is  called  the  abscissa,  instead  of  PE.  PD  and  AD  taken  together  are 
called  the  co-ordinates  of  the  point  P. 

Sen.  2.— The  pupil  will  see  that  this  device  for  locating  points  is  not 
unlike  the  method  of  locating  places  on  the  earth's  surface  by  means  of 
latitude  and  longitude. 

12.  Abscissas  are  represented  in  the  notation  by  the  letter  xt 
and  ordinates  by  y. 

13.  The    four    angles  into  which  the  plane    is  divided  by  the 
axes  are  distinguished  thus  :  The   angle  above  the  axis  of  abscissas 
and  at  the  right  of  the  axis  of  ordinates  is  called  the  First  Angle ; 
and  the  numbering  proceeds  from  right  to  left.     YAX   is  the  First 
Angle,  YAX'  is  the  Second,  X'AY'  is  the    Third,  and  XAY'  is  the 
Fourth. 

14.  In  order  to  indicate  in  which  of  the  four  angles  a  point  is 
located,  the  signs  -f-  and  —  are  used  on  the  following  principles  : 
abscissas  reckoned  from  the  origin  to  the  right  are  marked  +>  and 
those  reckoned  to  the  left  are  —  ;  ordinates  reckoned  upward  from 
the  axis  of  abscissas  are  +,  and  those  reckoned  downwards  are  — . 
Accordingly,  the  abscissas  of  points  in  the  1st  and  4th  angles,  as  A  D 
and  AD"'  are  -f,  while  those  in  the  2nd  and  3rd  angles  as,   AD'  and 
AD",  are  — .     Ordinates  in  the  1st  and  2nd  angles,  as  PD  and  P'D', 
are  +,  and  those  in  the  3rd  and  4th,  as  P"D"and  P'"D'"  are — . 


DEFINITIONS  AND   FUNDAMENTAL   NOTIONS.  9 

13.  The  quantities  used  in  General  Geometry  are  distinguished 
as  Constant  and  Variable. 

16.  A  constant  quantity  is  one  which  maintains  the  same 
value  throughout  the  same  discussion,  and  is  represented  in  the  no 
tation  by  one  of  the  leading  letters  of  the  alphabet. 

J?7.   Variable  quantities  are  such  as  may  assume  in  the  same 
discussion   any  value,  within   certain  limits   determined  by  the  na-' 
ture   of  the  problem,*   and  are  represented  by   the  final  letters  of 
the  alphabet. 

ILL.— In  Fig.  5  let  BECF   be  a  circle  whose  radius  is 
R,   and  XX'  and  YY"  be  the  axes  of  reference.    Kepre- 
Bent  the  abscissa  of  any  point  in  this  circumference  by  x 
and  the  corresponding  ordinate  by  y  ;  so  that  when  x 
signifies  AD,  y  shall  represent  PD  ;    when   —  x  is 
AD',  t/shallbe  P'D'  ;  when  — x  is  AD",  — t/shall 
be  P"D",  etc.     Now,  suppose   it  possible   to  represent 
the  relation  between  x,   y  and  R  by  an  equation  so  gen 
eral  as  to  be  true  for  all  points  in  this  circumference,  as 
P,  P',  P",  etc.     (It  will  subsequently   appear  that  this 
equation  is  x-  -j-  y-  =  R2.)     In  such  an  equation  R  would 

be  constant,  for  it  remains  the  same  for  all  positions  of  the  point  P  ;  and  a;  and  y 
would  be  variables,  since  they  vary  in  value  with  every  change  of  the  position  of 
P.  In  such  a  problem  it  is  evident  that  x  or  y  could  not  exceed  R,  hence  these 
variables  could  have  all  values  between  the  limits  of  -f-  R  and  —  R. 

SCH.  1. — Care  should  be  taken  not  to  confound  the  terms  constant  and 
variable  as  here  used,  with  known  and  unknown  as  used  in  algebra  : 
especially  as  the  notation  would  suggest  an  identity  which  does  not  exist. 
Both  the  known  and  unknown  quantities  of  Algebra  are  constants  ;  moreover 
the  constants  in  General  Geometry  may  be  either  known  or  unknown;  and 
the  same,  in  a  certain  sense,  may  be  said  of  the  variables. 

SCH.  2.  — In  order  that  the  variables  may  retain  their  peculiar  charac 
teristic,  we  cannot  have  as  many  equations  arising  from  a  particular  prob 
lem  as  there  are  variables  ;  thus,  if  there  are  two  variables  involved,  we  have 
but  one  equation.  In  algebra  such  problems  are  called  indeterminate,  since 
the  equation  does  not  determine  definite  values  of  the  unknown  quantities, 
but  can  be  satisfied  by  an  infinite  variety  of  values .  From  this  feature  of 
the  General  Geometry  it  is  sometimes  called  Indeterminate  Analysis.  The 
Calculus  is  also  embraced  under  the  same  term,  as  its  problems  involve  a 
like  feature. 


*  Our  limits  do  not  permit  a  discussion  of  the  continuity  of  functions  and  the  general  geo 
metrical  interpretation  of  imaginary  co-ordinates,  and  hence  for  simplicity  we  retain  the  con 
ception  of  imaginaries  as  impossible  quantities. 


10 


THE  CARTESIAN  METHOD  OF  CO-ORDINATES. 


18»  To  construct  an  equation,  or  find  its  locus,  is  to  draw  the  geo 
metrical  figure  represented  by  it. 


SECTION  II. 
Constructing  Equations,  or  Finding  their  Loci, 

10.  A  curve  is  continuous  when  its  course  is  uninterrupted 
both  in  extent  and  in  the  character  of  its  curvature. 


ILL. — A  circle,  an  ellipse,  and  the  curve  in  fig.  6  are 
examples  of  curves  continuous  in  extent  and  curvature. 
They  may  be  traced  throughout  by  the  uninterrupted 
movement  of  a  point.  The  curve  Fig.  7,  is  discontinuous 
in  extent ;  and  in  Fig.  8,  we  have  an  example  of  a  curve 
discontinuous  in  curvature.  Fig .  9  affords  an  example  of 
discontinuity  both  in  extent  and  curvature. 

20.  A.  Branch  is  a  continuous  portion  of 
a  curve.     In  Figs.  1  and  8  the  curves  have  two 
branches   each.       la   Fig.    9   there    are  four 
branches. 

21.  A  curve  is  symmetrical  with  respect 
to   either    axis,   or   to  any   line,  when  it   has 
the  same  form  on  both  sides  of  the  line,  or 
when  every  point  on  one  side  of  the  line  has  a 
corresponding  point  on  the  other.     The  curves 
in  Figs.  6,  7   and  9  are  symmetrical  with  re 
spect  to  the  axis  of  abscissas,  and  the  first  two 
with  respect  to  both  axes.     The  curve  in  Fig.  8 
is  not  symmetrical  with  respect  to  any  line. 


FIG.  6. 


FIG.  7. 


FIG.  8 


FIG.  9. 


22.    Prob.  To  locate  a  Point  whose  co-ordinates  are  given. 

SOLUTION. — Lay  off  from  the  origin,  on  the  axis  of  abscissas,  a  distance  equal  to 
the  given  abscissa,  to  the  right  if  the  abscissa  is  -f->  and  to  the  left  if  it  is  — . 
Through  the  point  thus  found  draw  a  line  parallel  to  the  axis  of  ordinates,  and 
lay  off  on  it  a  distance  from  the  axis  of  abscissas  equal  to  the  given  ordinate,  above 
if  the  ordinate  is  -f->  and  below  if  it  is  — .  The  point  thus  found  will  be  the  one 
required. 


CONSTRUCTING  EQUATIONS. 


Ex.  1.  Locate  the  point  x  =  3,  y  =  —  5. 


p 


SOLUTION. — Draw  the  axes  XX' and  YY'.     Lay      x/        A 

off  A  B  =  3  to  the  right,  as  a-  is  -f,  and  draw  BO 
parallel  to  YY'.  Then  take  PB  =5  below  the  axis 
of  abscissas  as  y  is  — ,  and  P  is  the  point  required. 

y     Q 

Sen.— Points  are  usually  designated  by  mentioning  FIG.  10. 

simply  their  co-ordinates,  as  the  point  3,  —5,  for  the 
point  in  the  last  example.     The  abscissa  is  mentioned  first. 

Exs.  2  to  9.  Locate  —6,  2;  —5,  —7;  —3,  0  ;  0,  —  3  ;  0,  0  ;  5,  —  1- 
2,  0  ;  0,  4. 

QUERIES.— Where  are  points  situated  whose  abscissas  are  0?  Where  are  points 
situated  whose  ordinates  are  0  ?  What  are  the  co-ordinates  of  the  origin  ?  In  what 
line  are  3,  —2  ;  3,  5  ;  3,  0  ;  and  3,  4  situated  ? 


To  find  the  locus  of  an  equation  between  two  variables; 
i.  e.,  to  construct  the  equation. 

SOLUTION.— Solve  the  equation  with  respect  to  one  of  the  variables.  Then, 
since  the  equation  expresses  the  relation  between  the  co-ordinates  of  all  points  in 
the  locus,  substitute  for  the  other  variable  any  values  which  give  real  values  for  the 
first,  and  locate  the  points  thus  determined.  These  will  be  points  in  the  locus  ; 
and,  by  determining  a  sufficient  number,  the  locus  can  be  sketched  through  them. 

y     O/v» 

Ex.  1.  Construct  the  equation  - — : — -  =  1.  ' 

o 

SOLUTION.— Solving  the  equation  for  y,  we  have  y  =  2.r  -f-  3.     Now  attributing 
arbitary  values  to  x,  we  make  the  following  table  of  corresponding  values  : 
When  x  =  1,     y  =  5,    giving  the  point  1,  5  ; 
"    a;  =  2,     y  =  7,       "         "        "      2,  7  ; 
"    z  =  3,     t/  =  9,       "         "        ••      3,9; 

etc.,         etc.,         "         "        •«      etc. 

Noticing  that  all  positive  values  of  x  give  real,  positive,  and  single  values  to  y,  we 
discover  that  the  locus  has  but  one  branch  which  extends  to  the  right   of  the 
axis  of  ordinates,  extends  indefinitely,  and  lies  above  the  axis  of  abscissas. 
Again,  giving  negative  values  to  ar,  we  have 

When  x  =  —I,     y  =  1,    giving  the  point  — 1,  1  ; 

"     &  =  — 2,     t/  =  — 1,     "       "       ««       __2,  — 1  ; 
.«    x  =  —  3,     y  =  —  3,     "       «       «      —3, —3; 

and  for  all  subsequent  negative  values  of  x,  y  has  real,  negative,  and  single  values. 
Hence  we  learn  that  the  locus  has  a  single  branch  extending  indefinitely  in  the  third 
angle. 

If  we  make  y  =  0,  x  =  —  1 2  ;  wherTce  we  see  that  the  locus  cuts  the  axis  of  abscis 
sas  at  — Ij,  0.  If  we  make  x  =  0,  y  —  3  ;  and  hence  the  locus  cuts  the  axis  of 
ordinates  at  0,  3. 


12  THE   CARTESIAN   METHOD   OF    CO-ORDINATES. 

Finally,  locating  these  points,  as  in  Fig.  11,  we  find  that 
the  line  M  N  includes  all  the  points,  and  hence  conclude  v,      8'9/ 

that  it  is  the  required  locus. 

Sen.— If  any  other  values  be  attributed  to  x,  either 
integral  or  fractional,  positive  or  negative,  and 
the  corresponding  values  of  y  deduced,  the  points 
thus  determined  will  fall  in  the  line  M  N. 

s\. 

Ex.  2.   Find  the  locus  of   the    equation 
Jo?  +  2y  =  4. 

FIG.  11. 

Result.  A  straight  line  cutting  the  axis  of  abscissas  at  8,  0,  and 
the  axis  of  ordinates  at  0,  2. 

Ex.  3.  Find  the  locus  of  the  equation  %x  —  1  =      r       ^. 
Result.  A.  right  line  passing  through  0,  4,  and  3,  —  1. 

24.  Sen. — If  there  is  nothing  in  the  nature  of  the  equation  to  make  an 
other  course  preferable,  it  is  customary  to  solve  it  for'y,  finding  the  value  in 
terms  of  x,  and  constants.  If,  however,  the  equation  is  above  the  second 
degree  with  respect  to  either  of  the  variables,  it  is  expedient  to  solve  it  with 
reference  to  the  variable  which  is  least  involved.  Thus,  in  order  to  construct 
3x  —  y2  =  2y3  —  y  —  5,  we  solve  with  reference  to  x,  and  then  substitute 
arbitrary  values  for  y,  finding  the  corresponding  values  of  x. 

23-  DEF. — The  Independent  Variable  is  the  one  to  which 
we  assign  arbitrary  values,  usually  x.  The  other  is  called  the 
Dependent  Variable. 

This  distinction  is  made  simply  for  convenience,  and  is  not  founded 
in  any  difference  in  the  nature  of  the  variables  :  either  variable  may 
be  treated  as  the  independent  variable. 

26.  SCH. — There  are  certain  peculiarities  of  loci,  which  readily  appear 
from  the  form  of  the  equation.  These  should  always  be  noted.  Observing 
them  is  called  Discussing,  or  Interpreting  the  Equation.  The  following  are 
some  of  these  points  : 

1st.  The  Intersection  of  the  locus  with  the  Axes.  Where  the  locus  cuts  the 
axis  of  abscissas  y  =  0;  hence  substituting  this  value  (0)  of  y,  in  the  equation, 
and  finding  the  corresponding  value  or  values  of  x,  determines  the  intersec 
tions  with  the  axis  of  abscissas.  In  like  manner,  making  x  =  0,  and  finding 
the  corresponding  values  of  y,  determines  the  intersections  with  the  axis  of 
ordinates. 

2nd.  TJie  Limits  between  which  the  locus  is  comprised,  and  its  continuity  or 
discontinuity  between  these  limits.  These  Questions  are  to  be  determined  with 
respect  to  each  axis.  The  limits  are  discovered  by  determining  the  great 
est  and  least  values  of  the  independent  variable  which  give  real  values  to  the 


CONSTRUCTING  EQUATIONS.  13 

dependent  one.  If  all  values  of  the  independent  variable  between  the  limits 
observed  in  this  way,  give  real  values  for  the  dependent  variable,  the  locus 
is  continuous  in  extent  between  these  limits.  If,  on  the  other  hand,  there 
are  certain  values  of  the  former  which  render  the  latter  imaginary,  the  locus 
is  discontinuous;  and  the  limits  of  discontinuity  are  to  be  observed  by  find 
ing  the  limits  between  which  the  values  of  the  dependent  variable  are  ima 
ginary. 

3rd.  Whether  the  locus  is  symmetrical  with  respect  to  an  axis,  or  with  any 
line,  or  not.  The  manner  of  determining  this  is  as  follows  :  If,  for  each  real 
value  of  one  variable,  the  other  has  two  values,  numerically  equal  but  with 
contrary  signs,  there  are  points  similarly  situated  on  opposite  sides  of  the  axis 
from  which  the  variable  having  two  values  is  reckoned,  and  hence  the 
locus  is  symmetrical  with  respect  to  that  axis.  Again,  if  there  is  any  line  so 
situated  that  the  values  of  the  intercepts  of  either  of  the  co-ordinates  be 
tween  it  and  the  locus,  on  both  sides  of  the  line,  are  equal,  the  locus  is 
symmetrical  with  respect  to  that  line. 

[NOTE.— There  are  many  other  characteristic  features  of  loci  which  appear  more  or  less  immedi 
ately  from  the  form  of  the  equation,  and  some  of  which  will  be  noticed  iu  a  subsequent  part  of  the 
course.  Those  now  mentioned  are  sufficient  for  our  present  purpose  if  the  pupil  becomes  perfectly 
familiar  with  them.  This  familiarity  can  be  attained  only  by  a  careful  study  of  examples.  In  fact 
it  is  hardly  probable  that  the  pupil  can  understand  the  full  purport  of  the  language  of  the  last 
scholium  until  he  has  solved  several  examples.  After  studying  a  few  which  follow,  he  can  return 
and  read  the  scholium  again,  and  be  bettor  able  to  see  its  meaning.] 

Ex.  4.  Find  the  locus  of  the  equation  x*  +  y*  =  25. 


SOLUTION,  y  =  ±  ^25  —  £2.  For  x  =  0,  y  =  5  and  — 5.  Hence  the  locus 
cuts  the  axis  of  ordinates  at  (0,  5)  and  (0,  — 5).  For  y  =  0,  x  =  5  and  —  5» 
Hence  the  locus  cuts  the  axis  of  abscissas  at  (5,  0)  and  ( — 5,  0).  Again,  as  every 
value  of  x  between  -f-  5  and  —5,  gives  two  real  values  for  y,  numerically  equal,  but 
with  opposite  signs,  the  locus  is  symmetrical  with  respect  to  the  axis  of  abscissas, 
and  continuous  between  these  limits.  In  like  manner,  x  —  -b  >/25  —  y-  shows 
that  the  locus  is  symmetrical  with  respect  to  the  axis  Of  ordinates,  and  continuous 
between  y  =  5,  and  y  =  —  5.  When  x  is  numerically  greater  than  5  (either  -)-  or 

)?  the  values  of  y  become  imaginary.  Hence  the  locus  is  comprised  between  the 

limits  x  =  5,  and  x  =  —  5.     From  x  =  =fc  -s/25  —  y-,  it  Appears,  in  like  man 
ner,  that  the  limits  in  the  direction  of  the  axis  of  ordinates  are  y  =  5,  and  —  5. 

Now  assigning  to  a;  arbitrary  values  between  -j-  5  and  —  5,  we  find  the  following 
table  of  values,  and  points  in  the  locus  : 
Whenic  =  l,  y=  =h  V*f24  =  =b4.9  nearly  ;  and  we  have  points  (1,  4. 9)  and  (1, — 4.9); 

<«    x=2,  y  =  ±  ^21  =±4.6  nearly  ;       "     "       "       "     (2,  4. 6)  and  (2,— 4.6;; 

«,    jc^S,  y—  ±-s/l6  =  ±4  "     "      "       "       (3,  4,  i  and  (3,— 4); 

"    x  =  4,  y=±-s/9=±3  "     "       "       "      (4,  3,)  and  (4, —3); 

For  negative  values  of  x  the  following  points  are  found  (— 1,  4.9)  and  (—  1,— 4.9); 
(—  2,  4.6)  and  (—2,  —4.6);  (—3,  4)  and  (—3,  —4);  (-4,  3)  and  (-4,  —  3). 


THE   CARTESIAN  METHOD   OF   CO-OllDINATES. 


Constructing  the  points  thus  determined 
they  are  found  to  be  in  the  circumference  of  a 
circle  whose  radius  is  5,  and  which  is  sym 
metrical  with  the  axes,  as  in  Fig.  12.  It  is 
also  to  be  observed  that  any  values  of  x, 
fractional  as  well  as  integral,  between  the 
limits  x  =  5,  and  x  =  —  5,  give  values  for  y 
which  locate  points  in  the  same  circumference. 


Ex.  5.  Construct  and  discuss 
tion  9«  +  4#2  =  3G. 


the 


equa 


SOLUTION. — Solving    the  equation    for  y,   y  •. 
d=  f  N/9  —  x2.  We  now  observe  that  for  x  =  0,  y  = 

±  2 ;  therefore  the  locus  cuts  the  axis  of  ordinates  at  (0,  2)  and  (0,—  2).  In  like  manner, 
making  y  =  0,  x  =  ±  3  ;  and  hence  the  locus  cuts  the  axis  of  abscissas  at  (3,  0)  and 
( —  3,  0).  Again,  for  each  value  of  x  which  renders  9  —  a2  >>  0,  i.  e. ,  for  each  value  be 
tween  x  =  3,  and  —  3,  y  is  real  and  has  two  values,  numerically  equal,  but  with  con 
trary  signs  ;  therefore  the  locus  is  symmetrical  with  reference  to  the  axis  of  abscis 
sas,  and  continuous  between  the  limits  x  =  3,  and  a;  =  —  3.  Beyond  these  values 
of  x,  y  becomes  imaginary,  and  the  locus  is  entirely  comprised  within  a-,  =  3  and 
x  =  — 3  along  the  axis  of  abscissas.  In  a  similar  manner  from  x  =  =b  f  <\/4  —  y~~, 
it  appears  that  the  locus  is  comprised  between  y  =  2  and  y  =  —  2,  and  is  sym 
metrical  and  continuous  with  respect  to  the  axis  of  ordinates. 

Finding  the  values  of  y  corresponding  to  a  sufficient  number  of  arbitrarily  taken 
values  of  x,  so  as  to  enable  me  to  sketch  the  curve,  we  have  the  following  table  of 
values  : 

For  x  =  0,      y  =  d=  2,  giving  the  points  «,  a'  in  f  ig.   13  ; 

"    x=    .5,    y  =  if  1.97,"       "         "      b,  V    "         " 

«    «  =  !,       y  =  ±  1.89,  "       «         «      c,  c'    " 

"    jr.  =  1.5,     j/==dbl.73,  «       "         "      d,  d'    " 

"    iC  =  2,       y=±1.49,  "       "         "      e,  e'    "         " 

"    £C=  2.5,    y=  ±  1.1     "       "         "      /,/    " 

"    x  =  2.75,  y=  db     .8,    "       "         "      ; 

"    x=  2.9,    y=db     .51,  "       " 

"    x—  3,       y  =  0 

Since  the  equation  contains  only  the  square  of  x,  neg 
ative  values  of  x  give  the  same  values  for  y  as 
positive  values  do,  and  the  portion  of  the  curve 
on  the  left  of  the  axis  of  ordinates  is    sym 
metrical  with  that  on  the  right. 


Finally,    locating    the    points,    as    made 
known  in  the   table,    and   a  similar  set  of 
points  on  the  left  of  the  axis  of  ordinates,  we 
have  an  ellipse  whose  axes  are  6  and  4,   Fig.  13. 

Ex.  6.  What  is  the  locus  of  v2  =  2x  —  6  ? 


CONSTRUCTING  EQUATIONS. 


15 


Ans. — It  cuts  the  axis  of  abscissas  at 
3,  0,  and  lies  wholly  to  the  right  of  this 
point,  extending  indefinitely  in  two 
branches,  one  above  the  axis  of  abscissas 
and  one  below  it ;  and  the  two  are  sym 
metrical  with  this  axis.  Fig.  14.  The 
branch  B  M  extends  indefinitely  in  the 
1st  angle,  and  B  M '  in  the  fourth.  The 
locus  is  known  as  a  Parabola. 


FIG.  14. 


Exs.  7  to  10.    Construct  the  following  equations  :    %x  -f   2y  =  4  ; 
2x  -f  3y  =  0  ;  &r*  +  5y*  =  12  ;  y*  —  6y  +  x*  =  16. 

Ex.  11.  Find  the  locus  of  &  —  y*  =  10. 

Ans.  —  The  locus  is  represented  in  Fig. 
15.  It  is  discontinuous  between  x  ==. 
v/lO  and  x  =  —  v/10  ;  but  to  the  right 
and  left  of  these  points,  it  extends  indefi 
nitely.  It  is  symmetrical  with  respect 
to  both  axes.  The  curve  is  known  as 
as  an  Hyperbola. 


FIG.  15. 


Exs.  12  to  23.  Construct  and  discuss  the  following  :  yz  — 

16  —  x3  ;  if-  =  lOx  —  xz  ;  y*  =  I2x  ;  x*  —  6x  +  9  + 
y*  4-  IQy  =  0  ;  25(y  +  4)»  +  16(*  —  5)»  =  400  ;  y* 
=  4  +  2  (x  —  3)'  ;  ?/2  =  #2  —  4  ;  y*  =  &c»  —  #3  +  5  ; 
#y  =  16  ;  y2  =  #2  —  #4 ;  y*  =  x*  —  a?2 ;  y2  —  x4  —  x3. 

Ex.  24.  Construct  and  discuss  the  equation  x  ==  log  y. 

Results. — Assuming  x  as  given  in  the  following  table  of 
values  (any  convenient  values  of  x  maybe  taken),  the 
values  of  y  can  be  found  from  a  table  of  logarithms. 


For  x=  0,  y  =  1. 
"  x  =  .2,  y  =  1.58  nearly. 
»  x=  .4,  y  =  2.51       " 
"  x  =  .6,  t/  =  3.98 


aj=  1,  y  =  10. 
etc.     etc.     etc.     etc. 


For  x  =  —  .1, 


.  8  nearly. 


=—  .22,  t/=. 
=  -A,  y= 
=  —  .7,  j/  =  . 


N 


IM 


'  x  =  —    1,  y  —  .l     "        x  A 

<   x  =  —    2,  y=.01    " 
etc.     etc.    etc.     etc. 

FIG.  16. 

Locating  these  values,  we  have  the  curve  MN,  Fig.  16,  which  is  called  the 
Logarithmic  Curve.  It  lies  wholly  above  the  axis  of  abscissas,  as  negative  numbers 
have  no  logarithms.  It  extends  on  both  sides  of  the  axis  of  ordinates,  and  cuts  it 
at  (0,  1,)  a  point  through  which  all  logarithmic  curves  pass,  in  whatever  system  the 


16 


THE  CARTESIAN  METHOD   OF   CO-OEDINATES. 


logarithms  be  taken,  since  log  1  =  0  in  all  systems.     The  curve  extends  inde 
finitely  to  the  right  and  to  the  left  ;  but  the  portions  are  not  symmetrical. 

Ex.  25.  Construct  x  =  log  y,  assuming  2  as  the  base  of  the  system 
of  logarithms  ;  giving1  y  =  2* . 

The  values  are,  x=  0,  y  =  1 ;  a?  =  1,  y  =  2;  x=2,  y  =  4;  #  =  3, 
y  =  8;  etc.  Also,  x  =  — 1,  y  =  .5  ;  or  =  —2,  y  =  .25  ;  x  — 

— 3,  y  =  i;  #  =  —4  y  =  -rV> etc- 

QUERIES. — Locating  this  curve  on  the  same  axes  with  the  preceding,  what  common 
point  do  they  possess  ?  Does  the  right  hand  branch  of  this  lie  to  the  right,  or  to 
the  left  of  the  former  ?  Does  the  left  hand  branch  approach  the  axis  of  abscissas 
more  rapidly,  or  less  rapidly  in  the  latter  than  in  the  former?  What  makes  these 
differences  ?  How  would  it  be  with  a  base  100  ? 

Ex.  26.   Construct  and  discuss  y  =  sin  x. 

SUGS.  — The  unit  arc  is  a  portion  of  the  circumference  equal  to  the  radius.  This 
arc  is  57.3°  nearly  ;  since,  radius  being  unity,  the  semi-circumference  is  3.1416,  and 

=  57.3°  nearly.     Hence  the  following  table. 


3.UK5 

For  x  =    0°  =     0, 


"  x.  =  300  —  .52, 
"  x  =  403  =.70, 
»  x  =  50^  =  .87, 
etc.  etc.  etc. 


=  .50 


y  =  -77 

etc. 


d  the  &n-        \ 
it  cut  the 
Is  it  lim-    . 


This  curve  is  called  the  Sin 
usoid.      Where  does 
axis   of  abscissas? 
itecl?     What  are  the    limits  of 
y  ?     What  is  the  meaning  of  x 
=  — 1(P,  x  =  —203,  etc.? 

Exs.  27  to  33.  Construct  y  = 
tan  x  ;  y  =  cot  x  ;     y  =  cos  x  ; 
y  =  versino? ;    y  =  coversin# 
y  =  sec  x  ;  y  =  cosec  x. 

Son.  — These  loci  can  be  con 
structed  with  sufficient  accura 
cy  without  the  numerical  com 
putations.      Thus,   taking    the 
Ex.  ?/  =   tan.r,  draw  a  circle  ON,  Fig. 
IS,  with  any  convenient  radius.     Divide 
a  quadrant,  as  M  N,  into  equal  parts,  each 
so  small   that  for  practical  purposes  the 
chord  and  arc  may  be  considered  equal. 


For  x  =  180°  =3.14, 
«  x  =  1900  =  3.31, 
"  x  =  200°  =  3.49, 
"  x  —  2100  =  3.66, 
•«  x  =  220^=3.84, 
"  x  =  2300  =  4.01, 
etc,  etc.  etc. 
Y 


0 

.i7 

.34 

.50 
.64 
.77 

etc. 


Fia.  18. 


THE   POINT  IN  A  PLANE.  17 

Estimating  the  tangents  and  arcs  from  M,  and  having  drawn  the  tangents  as 
in  the  figure,  lay  off  the  arcs  on  the  axis  of  abscissas.  At  the  extremity 
of  Al  lay  off  an  ordinate  equal  to  tangent  Ml,  etc.,  etc.  There  are  an. 
infinite  number  of  similar  infinite  branches  to  this  curve. 

On  the  figure  used  for  getting  the  tangents,  when  the  arc  passes  90°  the 
tangents  (and  hence  the  ordinates)  become  negative.  Strictly  speaking, 
negative  values  of  x  would  be  obtained  by  measuring  the  arcs  on  the  circle 
from  M  downward,  or  from  left  to  right ;  so  that,  from  x  =  0  to  x  = 
— 90°,  the  tangents  (and  hence  the  ordinates)  are  negative.  From  x  =  — 90° 
to  x  =  — 180°,  the  tangents  (and  hence  the  ordinates)  are  positive.  Where 
do  the  branches  cut  the  axis  of  abscissas  ?  At  what  values  of  x  do  the 
ordinates  become  infinite  ? 


SECTION  III. 
The  Point  in  a  Plane, 

27.   BET. — The  Equations  of  a  Point  are  the  algebraic 
expressions  which  determine  its  position. 


Dr 


D 


28.  Prop.  The  Equations  of  a  Point  in  a  plane  are  x  =  a,  and 
y  =  b,  in  which  the  signs  of  a  and  b  are  general. 

DEM.— If,  as  in  Fig.  19,  we  make  AB  —'a,  and 
through  B  draw  DE  parallel  to  YY',  every  point 
in  D  E  will  have  its  abscissa  equal  to  a.      In  like          _ — - 
manner  make  AC  =  6,  and  draw  FG  parallel  to 
XX', and  every  point  in  FG  will  have  b  for  its     X1      B 
ordinate.      Hence  the  point  P  has  a  for  its  abscissa, 
and  Z>  for  its  ordinate  ;  and  since  two  straight  lines, 
can  meet  in  only   one  point,    P  is  the  only  point' 
which  has  these  co-ordinates.    Therefore  x  =  a,  and 
y  =  6,  determine  the  position  of  a  point.     Q.  B.  D.  FIG   19 

Sen.  1.— If  we  have  x  =  —a,  and  y  =  bfP'  is  the  point.  If  x  =  —a,  and 
y  =  — &,  P"  is  the  point,  etc. 

Sen.  2.— If  x  =  a  =  0,  and  y  =  b,  the  point  is  in  the  axis  of  ordinates. 
If  x  =  a,  and  y  =  b  =  0,  the  point  is  in  the  axis  of  abscissas,  x  =  0, 
y  =  0  characterizes  the  origin. 

Sen.  3.— A  point  is  usually  designated  by  simply  naming  its  co-ordinates, 
the  abscissa  being  mentioned  first.  Thus  the  point  (m,  n}  is  the  same  as 
the  point  x  =  m,  and  y  =  n. 

Exs.  1   to  6.      Locate  the  points  x  =  —3,  y  =  4 ;  (5,   7)  • 

(0,  -5);  (0,4);  (0,0);  (6,0). 


18  THE   CARTESIAN   METHOD   OF  CO-ORDINATES. 

Exs.  7  to  10.     How  are  the  points  (5,  $) ;  ( J,  —6) ;  ( J,  m) ;  (—  n,  J) 
situated  ? 

Answer  to  the  first. — In  a  line  parallel  to  the  axis  of  ordinates  and  at 
a  distance  5  from  it.  Any  point  in  this  line  fulfills  the  conditions, 
since  y  =  -£,  i.  e.,  is  indeterminate. 

Exs.  11,  12.  Construct  the  triangle  whose  Tertices  are  ( — 3,  4); 
(5,  — 1);  and  (2,  — 6).  Also  the  triangle  whose  vertices  are  (0,  3); 
(—5,0);  and  (0,0). 

Exs.  13,  14.  What  figure  is  that  the  vertices  of  whose  angles  are 
(2,  3);  (2,  8);  (7,  8);  and  (7,  3)?  What  figure  is  that  the  vertices  of 
whose  angles  are  (2,  9);  (—8,  9);  (—8,  1);  and  (2,  —1)? 


20.    Prop.     The    Distance    between    two   points    in   a   plane   is 
(x'  —  #'')*+  (y'  —  y"Y,  in  which  (x,  y't)  and  (#",  y")  are  the  points. 


DEM. — Let  the  points  (xr,  y' )  and  (x"  t/")be  represented 
by  P'  and  P",  as  in  Fig.  20,  and  the  distance  between 
them,  P  P",  by  D.  Draw  P"D  parallel  to  AX. 
Then  P"D  =  x'  —  x",  and  P  D  =  y'  —  y".  From 
the  right  angled  triangle  P'  P"  D,  we  have, 

D  =  vV  —  a";3  4-  (y'  —  y")--    o-  »•  »• 


B       C 
Y' 

FIG.  20. 


COK. — If  either  of  the  points,  as  P",  is  at  the  origin,  its  co-ordinates 
are  0,  0,  and  D  =  \/x'*  -j-  y'*. 

QUERIES.—  When  P"  is  in  the  axis  of  abscissas  and  at  the  right  of  the  origin, 
what  is  the  formula?  The  same  with  P"  at  the  left  of  the  origin,  give  D  = 
«/(x  +  x")>2_j_yx  If  P'  is  in  the  1st  angle  and  P"  in  the  3rd,  what  is  the 
formula?  If  P' is  in  the  axis  of  abscissas  and  P"  in  the  axis  of  ordinates?  If 
one  is  in  the  2nd  and  the  other  in  the  4th  angle  ? 

Sen. — Observe  that  the  formula  D«=  */(x  —  x"}~  +  (y' y")2  is  strictly 

general,  only  noticing  carefully  the  effect  of  the  position  of  the  points,  upon 
the  signs  of  their  co-ordinates.  Thus  for  a  point  P",  in  the  4th  angle,  we 
have  x",  and  —  y";  which,  substituted  in  the  formula,  gives  for  P'  in  the  1st 
angle  and  P"  in  the  4th,  D  =-.  \/ (x  —  x")*-f~d/'  +  y"~7*. 

EXAMPLES.— Find  the  distances  between  the  following  points  taken 
two  and  two:  (3,  5);  (2,  6);  (—3,  —2);  (—1,  4);  (—2,  — 1); 
(—5,— 7);  (-3,0);  (0,  — 4);  (0,0);  (-5,0). 


THE  EIGHT  LINE  IN  A  PLANE.  19 

SUCTION  IV. 

The  Eight  Line  in  a  Plane, 

30.  Vw.—TJie  Equation  of  a  Locus  is  an  equation  which 
expresses  the  relation  between  the  co-ordinates  of  every  point  in  the 
line. 

31.  Prop.    The  Equation  of  a  Eight  Line  passing  through  two  given 

points  is  j—y'  =  ^^—r  (x  •—*'),  in  which  (*>  j)  is  any  point  in  the 
x  —  x 

line,  and  (x',  y')  and  (x",  y")  are  the  given  points. 


DEM.—  Let  M  N  be  any  right  line  referred  to  the  rect- 
angular  axes  XX',    YY'.     Let  P  be  any  point  in  the 
line,  and  designate  its  co-ordinates,  AD  and  PD,  by  a 
andy.     Let  P'  and  P"  be  the  given  points  whose 
co-ordinates  are  x',y',  and  x",  y"  ,  respectively. 
Now  drawing    P'Eand  P"  F  parallel  to    AX, 
the  triangles  P  E  P'  and  P'  F  P"  are  similar,  and 
givePE:P'F::P'E:P"F.  But  PE  =  y-y'> 
P'F  =y'  —  y",  P'E  ==  x  —  x',    and   P"  F  =  FIG.  21. 

x'  —  x"  ;  hence,  substituting  these  values,  we  have 

y  _y'   :  y'—y"  -.  :  x  —  x  :  x'  —  x",  or  y  —  y'  =  ^~gr  («"—  «')•     Q- 
COE.     1.—  Since    P'P"F  ==  NGX,    and   |7—  -|-=—  - 


P/N 


'• 


B     C     D  X 
Vr 


tan  P'P'T,  we  have  ^7—  —  ,r  —the  tangent  of  the  angle  which  the  line 
makes  with  the  axis  of  abscissas. 

Tf   «./  _  «.//  y        y  .  —  y        •*    =  oo.  which  bemsr  the  tangent  of 
'  xi  _  x"  0 

90°,  shows  that  the  line  is  perpendicular  to  the  axis  of  abscissas. 
This  is  as  it  should  be,  since  if  x'  =  x",  the  points  P'  and  P"  are 
equally  distant  from  YY',  and  hence  M  N  is  perpendicular  to  XX'. 

Tf  „'   -       «/"     ??Lllj£'  =  -  =  0,  which  being  the  tangent  of  0°, 
"    y  *    xi  _  x"       x'  —  x" 

shows  that  the  line  is  parallel  to  (makes  no  angle  with)  the  axis  of 
abscissas.  This  is  as  it  should  be,  since  by  observing  the  figure,  it 
appears,  that  when  y'  =  y",  M  N  is  parallel  to  XX'. 

32    COR   *—The  Equation  of  a  Might  Line  passing 
through  one  given  point.    If  *'  =*",  and  </  =  </",  we  have 


20 


THE   CARTESIAN   KETHOD   OF   CO-ORDINATES. 


y  —  y'  =  -J  (x  —  x'),  and,  by  putting  the  indeterminate  expression, 
'£,  =  a,  l/  —  y'  =  a(x  —  x'}.  This  is  the  equation  of  a  straight  line 
passing  through  a  given  point,  since  the  conditions,  x'  =  x",  y'  =  y", 
make  P'  and  P"  coincide.  The  a  is  indeterminate,  as  it  should  be, 
since,  through  one  given  point,  an  indefinite  number  of  straight  lines 
can  be  drawn. 

33.  COR.  3. — TJie  Common  Equation  of  a  Right  Line. 

If  in  y  —  y'  —  a(x  —  x')y  we  make  x'  =  0,  and  designate  the  corres 
ponding  value  of  y'  by  b,  so  that  the  given  point  shall  be  the  point  in 
which  the  line  cuts  the  axis  of  ordinates,  we  have,  after  reduction, 
y  =  ax  +  b,  which  is  the  common  equation  of  the  straight  line.  In  this 
equation  a  is  the  tangent  of  the  angle  which  the  line  makes  with  the 
axis  of  abscissas,  and  b  the  distance  from  the  origin  to  where  the  line 
intersects  the  axis  of  ordinates. 

Sen.  1. — Discussion  of  the  Equation  y  =  ax+b.  If  b  be  -f>  the  line 
cuts  the  axis  of  ordinates  above  the  origin  ;  if  — ,  below  ;  if  0,  at  the  origin. 
lu  the  latter  case,  we  have  y  =  ax,  as  the  equation  of  a  right  I  hie  passing 
through  the  origin.  If  a  be  -f,  the  line  makes  an  acute  angle  with  the  axis 
of  abscissas,  (i.  e.,  it  inclines  to  the  right,  as  tko 
lines  in  Fig.  22),  the  tangent  of  an  acute  angle 
being  -J-.  If  a  be — ,  the  angle  is  obtuse,  (i.  <?., 
the  line  inclines  to  the  left,  as  in  Fig.  23),  since  the 
tangent  of  an  obtuse  angle  is  — .  If  a  •=  0  the 
line  is  parallel  to  the  axis  of  abscissas,  and  if  a  = 
oo ,  it  is  perpendicular,  as  will  readily  appear. 


=5? 


b  for  x,  we  have 
the 


Sen  2. — If  we  solve  y  =  ctx 
x  —-  y  —  -,  in  which  —  is  the  tangent  of 

angle   which  the  line  makes  with  the    axis    of 
ordinates,  since,  in  Fig.  21,  the  angle  AHG  = 


V 

FIG.  22. 


NHY  =90°  —    NGX 


In  this 


form,  - 
a 


the  distance  on  the  axis  of  abscissas  from  the 

origin  to  where  the  line  cuts  it  (AG),  since  the 

base  of  a  right-angled  triangle  is  equal  to  the  perpendicular  divided  by  the 

tangent  of  the  angle  at  the  base. 

34.  COR.  4.—T7ie  Equation  of  a  Right  Line  referred 
to  oblique  axes.    If  the  axes  are  oblique,  we  still  have  the  same 

f /'      ylt 

forms,  but  in  this  case  ~ -•— ,  or  a,  signifies  the  ratio  of  the  sines  of 


the  angles  which  the  line  makes  with  the  axes,  since  the  sines  of  the 


THE   EIGHT   LINE   IN   A   PLANE. 


21 


angles  of  a  plane  triangle  are  to  each 
other  as  the  sides  opposite.  Thus  in 
Fig.  2^,  P'F  (ory'—  y»):  P"F(ora?'-ar") 
:  :  AH  :  AG  !  :  sin  AGH  :  sin  AHG. 
Putting  ft  for  the  angle  included  by  the 
axes,  and  a  for  the  angle  which  the  line 
makes  with  the  axis  of  abscissas,  we  get 

Bin  a 


^x 


FIG.  24 


sn  a 

,,         .    .  ,  .,        ,  and,  finally,  y  =  -  —  r  ---  x  -f  6,  as  the  equa- 
—  x"        sm(fi  —  a)  Bin  (fj  —  a] 


?/'  —  y" 

—.  -  '—, 
x1  —  x 

tion  of  a  right  line  referred  to  oblique  axes. 
Ex.  1.  Construct  the  equation  y  =  2or  -f  3. 

SOLUTIONS.  —  There  are  three  methods  of  solution,  1st.  By  any  two  points.  -As 
it  is  known  to  be  an  equation  of.  a  right  Hue  from  its  form,  if  any  two  points  be 
determined,  as  in  the  last  section,  the  position  of  the  line  will  be  known.  For  ex 
ample,  for  x  =  3,  y  =  9,  and  for  x  =  —  2,  y  =  —  1  ;  whence,  locating  these  points 
and  drawing  a  line  through  them,  we  have  the  construction. 

2nd.  By  the  intersections  with  the  axes.  —  This  is  only  a  modification  of  the  1st 
method,  merely  making  y  =  0,  whence  x  =  —  1  A,  and  making  x  =  0,  whence 
y  =  3,  constructing  these  intersections,  and  passing  a  line  through  them.  (The 
pupil  should  execute  the  figures.) 

3rd.  By  means  of  the  tangent  of  the  angle  which  the  line  makes  ivith  the  axis  of  ab 
scissas.  Since  b  =  3,  we  may  lay  off  AC  =3  above  the  origin,  and  thus  determine 
C  as  a  point  in  the  line.  Through  C  drawing  C  E  parallel  to  AX  and  constructing 
the  angle  N  C  E  BO  that  its  tangent  shall  be  2  (by 
taking  CD  any  convenient  length  and  erecting 
the  perpendicular  FD  =2CD),  the  line  NM 
is  the  one  sought.  —  Or,  having  located  C,  take  AQ 


iAC,  whence  tan   AGC=- 

AG 


2,  will 


give  the  construction.  Or,  again,  drawing  any  line 
making  with  XX'  an  angle  whose  tangent  is  2,  and 
drawing  a  line  parallel  to  it  through  C,  the  latter 
will  be  the  line  sought. 


L 


FIQ.  25. 


Sen. — If  the  tangent  were  — ,  CD  would  be  laid  off  to  the  left  of  C,  or 
the  perpendicular  FD  let  fall  below  D. 

Ex.  2.  Produce  the  equation  of  a  line  passing  through  ( — 3,  5),  and 


SOLUTION. — Here  x'  =  — 3,  x"  =2,  y'  =  5  and  y"  =  — 1.     Now,  substituting 
these  values  in  t/  —  y'  =  —, ^  (x  —  x'),  and  reducing  to  the  form  y  =  ax  -f-  6, 

it      X 

we  have  y  =  —  1.2x  -f- 1.4.      The  pupil  should  construct  this  equation,  and  then 


22  THE   CARTESIAN   METHOD   OF   CO-ORDINATES. 

verify  the  result  by  locating  the  points  (—3,  5),  and  (2,— 1),  observing  that,  if  the 
work  is  correct,  they  will  fall  in  the  line.  Algebraically,  we  verify  the  result  by  sub 
stituting  in  the  equation  y  =  —  1.2#  -f- 1.4,  successively  for  x  and  y,  ( — 3,  5),  and 
(2,  <— 1),  each  of  which  must  satisfy  the  equation,  as  it  expresses  the  relation  between 
the  co-ordinates  of  any  point  in  the  line.  Substituting,  we  get  5  =  3.6  -f- 1.4, 
and  — 1  =  —  2.4  -f- 1.4,  both  of  which  are  correct. 

Ex.  3.  What  angle  does  the  line  which  passes  through  the  points 
(3,  5),  and  ( — 7,  2)  make  with  the  axis  of  abscissas?  • 

Ans.,  16°  42'  nearly.      > 

Ex.  4.  Produce  the  equation  of  a  line  passing  through  the  point 
(2,  — 3),  and  making  an  angle  with  the  axis  of  abscissas  whose  tan 
gent  is  4.  Ans.t  y  =  4:X  —  11. 

Ex.  5.  Produce  the  equation  of  a  line  passing  through  ( — 1,  0),  and 
( — 4,  — 5),  construct  by  the  3rd  method,  and  verify  the  equation  by 
locating  the  points. 

Ex.  6.  Construct  the  triangle  the  equations  of  whose  sides  are  y  = 
\x  +  3,  y  =  —  $x  +  4  and  y  =  \x  —  1. 

Ex.  7.  What  is  the  equation  of  a  line  which  cuts  the  axis  of  ordi- 
nates  at  3  above,  and  the  axis  of  abscissas  at  5  to  the  left  of  the  origin  ? 
(Notice  that  this  is  a  case  of  a  line  passing  through  two  points.) 

Ans.,  y  =  f  x  -f-  3. 

Ex.  8.  What  line  is  y  =  0.x  ?  What  is  x  =  0.  y  ?  How  is  y  = 
0.x  +  4  situated  ?  How  y  —  0.x  —  5  ? 

Ex.  9.  Find  the  angles  which  the  following  lines  make  with  the 
axis  of  abscissas  :  viz.,  the  line  passing  through  (3,  5),  and  ( — 1,  — 4); 
through  (5,  —2),  and  (5,  3);  through  (—3,  2),  and  (7,  2).  How  are 
these  lines  severally  situated  ? 


33.  Prop.  Every  Equation  of  the  First  Degree  between  two  variables 
is  an  equation  of  a  right  line. 

DEM. — Every  such  equation  may  be  put  in  the  form  Ay  -j-  .Bx  -f-  C  =  0,  in 
which  A  and  _B  are  the  collected  coefficients  of  y  and  x,  and  G  is  the  sum  of  the 

7?  C* 

absolute  terms.     By  transposition  and  division  we  have  y  = -r-x  -    — .  Now 

A  A 

7?  C* 

putting -j-  =  a  and r  =  6,  there  results  the  known  form  y  =  ax  -f  &•  Q-  E.  D.' 

^j.  ./I 

Son. — If  B  and  A  have  like  signs,  the  line  makes  an  obtuse  angle  with 
the  axis  of  abscissas  ;  and  if  they  have  unlike  signs,  it  makes  an  acute  angle. 
If  B  =  0  the  line  is  parallel  to  the  axis  of  abscissas,  and  if  A  =  0  it  is 
perpendicular.  If  A  and  C  have  like  signs,  the  line  cuts  the  axis  of  ordi- 
nates  below,  and,  if  unlike,  above  the  origin.  If  G  =  0  the  line  passes 


OF  PLANE  ANGLES,   AND  THE  INTERSECTION  OF  LINES. 


23 


through  the  origin.     In  general,  if  an  algebraic  equation  has  no   absolute 
term,  the  locus  passes  through  the  origin.     (Why  ?) 


Ex.  1.  Reduce 


to  the  form  Ay  -f 


3  ^ 

fix  +  G  =  0,  and  describe  the  line   according  to  the  suggestions  in 
the  preceding  scholium. 

Ans. — The  equation  is  y  —  13#  —  21  =  0.  A  =  1,  B  =  — 13, 
and  (7  =  — 21.  As  ^  and  B  have  unlike  signs  the  line  makes  an 
acute  angle  with  the  axis  of  abscissas,  the  tangent  of  which  is  13.  It 
cuts  the  axis  of  ordinates  above  the  origin  at  a  distance  of  21. 

2  —  y x  +  y  ^    r,  -\-  2?/ 


Ex.  3.  In  like  manner  discuss  3  — 


3 


G.y  +  y    x  —  y  _  337  +  ?/ 

~~~;  " 


Ex.     4.     Construct  the  figure   the  equations  of    whose    sides    are 
/  -f  2^;  =  3x  -f  3  -f  y  ;  -  —  1-^.  ==  2o;  —  G  —  y  ;  3?/  +  2.r  —  6 

-  -f  1 ;  and  x  +  ?/  =  —  3.    .What  is  the  figure  inclosed  ? 


SECTION  V. 


Of  Plane  Angles,  and  the  Intersection  of  Lines, 

36.   Prop.     The  expression  for  the  value  of   an   angle  included 

— — ,  in  which  V  is  the  angle  included 


between  two  lines  is  tan  V  = 


by  the  lines,  and  a  and  a'  are  the  tangents  of  the  angles  which  the  lines 
make  with  the  axis  of  abscissas. 

DEM.—  Let  MN  and  M'N',  Fig.  26,  be  two  lines 
•whose  equations  are  respectively  y  =  ax  4-  &  and 
y  =  ax  -f-  &'.  Now  C  BX  being  exterior  to  the  triangle 
BCD,  we  have  DCB  =  CBX  —  CDB,  or  by 
trigonometry 

tan  CBX  —  tan  CDB 


tan  D  C  B  = 


tan  C  BX   X  tan  C  D  B 


^-    X'    D, 


But  DCB=F,  tan CBX= a',  and  tan  CDB     IvT 


,,       a   —  a, 

a.      .  • .  tan  V  =  - — • ;.     Q.  E.  D. 

1        aa, 


Yl 

FIG.  26. 


24  THE   CAETESIAN   METHOD   OF   CO-ORDINATES. 

SCH. — In  applying  this  formula  to  any  particular  example,  we  may  obtain 
two  results,  numerically  equal,  but  with  opposite  signs.  Thus,  if  the  two 
lines  are  y  =  1x  -f  4,  and  y  =  3z  —  5,  and  we  let  a'  =  2,  and  a  =  3,  we 

2 Q  -. 

have   tan  V  =  —  —  =  —  --.     But,  if  we  let  a  =  2,  and  a'  =  3,  we  have 

3  —  2        1 
tan  V  =  2~7T~  £J  =  n-     This  ambiguity  is  as  it  should  be,  since  the  two  lines 

form,  in  general,  two  equal  acute,  and  two  equal  obtuse  angles  with  each 
other  ;  and  as  these  angles  are  supplements  of  each  other,  they  have  tan 
gents  numerically  equal  but  with  opposite  signs. 

37.  I*vob.     To  find  the  equation  of  a  line  which  makes  any  required 
angle  ivith  a  given  line. 

SOLUTION.— Let  y  =  ax  -f-  &  he  the  equation  of  the  given  line,  y  =  a'x  -f-  V  be 
that  of  the  required  line,  arid  ra  the  tangent  of  the  required  angle.  As  the  relative 
directions  of  the  hues  depend  solely  upon  a,  a',  and  m,  the  problem  consists  in 

finding  the  unknown  a',  in  terms  of  the  given  tangents  a  and  m.     But  m  =  • —        a 

i  -{-  aa 

,      ,,  ,.  ...  ,        CL  4-  in  a  -\-  m 

by  the  preceding  proposition;   whence  a  = ' ;  andv  — • x-4-6'is 

1  —  am  1  —  am 

the  equation  of  the  required  line. 

SCH. — In  this  form  b'  is  undetermined,  as  it  should  be,  since  there  may  be 
an  indefinite  number  of  lines  which  will  satisfy  the  condition,  all  having  the 
same  inclination  to  the  axis  of  abscissas,  but  cutting  the  axis  of  ordinates 
at  different  points. 

COR.  1. — If  the  required  line  is  to  pass  through  a  given  point  (x',  y'),  ice 

,  a  -f  m     , 

have  y  —  y    =   — (x  —  x'). 

1  —  ani 

38.  COR.  2. — If  the  required  line  is  to  be  parallel  to  the  given  line, 
m  =  0,  and  we  have  a'=  a.     The  equations  then  become  y  =  ax  +  b', 
and    y  —  y'  =  a(x  —  #'),   both   of    which    lines    are    parallel    to 
y  =  ax  -f  6. 

39.  COR.  3. — If  the  lines  are  to  be  perpendicular  to  each  other,  m  =  oo. 

•*•  a'= •  = *  = ,  or  1  +  «a'  =  0,  which  is  called  the 

1  —  am       — am  a 

equation  of  the  condition  of  perpendicularity.  The  equations  of  lines 
perpendicular  to  y  =  ax  -f-  b  will  therefore  be  y  == x  -f-  b',  and 

V  —  y'  = (%  —  *')>  *ne  ^tter  passing  through  (x',  y'). 

*  The  principle  upon  which  this  reduction  is  effected  is,  that  the  finite  terms  a  and  1  added  to  the 
infinites  ?n  and  — am  must  be  dropped.  The  axiom  is,  finites  added  to  infinites  do  not  (apprecia 
bly)  affect  the  ratio  of  the  infinites.  The  word  appreciably  is  thrown  ill  to  aid  the  student's 
apprehension.  It  is  not  required,  nor  is  it  strictly  correct. 


OF  PLANE   ANGLES,   AND   THE  INTERSECTION   OF  LINES.  25 

* 

Sen.  2.  —  Two  lines  are  parallel  to  each  other  when  the  two  equations  being 
reduced  to  the  form  y=ax-}-b,  the  coefficients  of  x  are  the  same  in  both  ; 
and  they  are  perpendicular  when  these  coefficients  are  reciprocals  of  each 
other  with  opposite  signs. 

Ex.  1.  Find  the  angle  included  between  y  =  —  x  -f  2,  and  y  =  3x  —  6. 

Result,   Tan  7=  ^t-\  =  —  2.     /.   The  angle  is  116°34'. 
JL  —  o 

Ex.  2.  What  are  the  angles  of  the  triangle  the  equations  of  whose 
»  sides  are  2?y  —  5  =  y  —  x  ;  y  -f  ±x=  8,  and  y  =  -J-a;  ? 

A        (    The  tangents  of  the  angles  are  .6,  —  21,  and  1.5. 
"  1    The  angles  are  nearly  30°58',  92°44',  and  56°19;. 

Ex.  3.  Write  the  equations  of  three  lines,  each  parallel  to  ?/  —  2#  — 
11,  and  construct  the  lines  therefrom. 

Ex.  4.  Write  the  equation  of  a  line  parallel  to  y  —  %x  —  5  and 
passing  through  (  —  6,  4).  Eeduce  the  equation  of  the  parallel  to  the 
form  y  =  ax  +  6,  and  then  construct  both  lines  from  their  equations. 
Verify  the  result  by  constructing  the  given  point  ;  also  by  observing 
that  the  coefficients  of  x  in  both  equations  are  equal,  and  that  the 
co-ordinates  of  the  given  point  satisfy  the  equation  of  the  parallel. 

Ex.  5.  Write  the  equation  of  a  line  passing  through  (  —  £,  -f  ),  and 
parallel  to  %x  —  %y  =  2.  Verify  as  in  Ex.  4. 

Result.  The  equation  isy—-  %x  -f 


Ex.  6.  Write  the  equations  of  three  lines  each  perpendicular  to 
\y  _  2#  =  1,  reduce  them  to  the  form  y  —  ax  +  6,  and  verify  the 
results  by  construction. 

Ex.  7.  Write  the  equation  of  a  line  perpendicular  to  2y  —  4  =  x, 
and  passing  through  (I,  —  3).  Verify  as  before. 

Ex.  8.  Write  the  equations  of  lines    perpendicular  to  -  —  —  —  = 

A 

^X  __  2,  and  severally  passing  through    (—2,3);    (0,  —  5);    (0,0); 
and  (—3,  0). 

Ex.  9.  What  is  the  angle  included  between  y  =  Q.x,  and  y  —  Zx 
—  5  ?  Between  x  =  O.y,  and  y  =  2r  -f  1  ? 

Ex.  10.  What  is  the  equation  of  a  line  passing  through  (  —  6,  0), 
and  perpendicular  to  y  =  O.a:  +  5?  Ans.,  x  =  O.t/  —  6. 


26  THE   CARTESIAN   METHOD   OF   CO-ORDINATES. 

Ex.  11.  "What  is  the  equation  of  a  line  passing  through  ( — 1,  3),  and 
making  an  angle  of  45°  with  y  =  2x  —  5  ?  Arts.,  y  —  — 3a?. 

Ex.  12.  Produce  the  equation  of  a  line  passing  through  ( — 4,  — 5) 
and  making  an  angle  of  71°  34'  with  y  =  — 2x  -f  7  ? 

Result,  y  =  fr  —  4f,  calling  tan  71°34',  =  3. 


40.  I*TOb.     To  find  the  point  or  points  of  intersection  of  two  lines. 

SOLUTION.  —  For  a  common  point  the  values  of  x  and  y  are  the  same  in  both 
equations,  and  only  for  such  a  point.  Therefore,  making  the  equations  simultaneous 
restricts  the  values  of  x  and  y  to  the  required  point  or  points.  Consequently,  w* 
have  only  to  solve  any  two  given  equations  for  the  values  of  x  and  y  in  order  to 
find  the  point  or  points  in  which  the  loci  intersect. 

Sen.  1.  —  The  general  formulae  for  the  value  of  the  co-ordinates  of  the 
point  of  intersection  of  two  straight  lines  whose  equations  are  y  =  ax  -f-  b, 

b'  —  b  ab'  —  a'b 

and  ii  =  a  x  -+-  b  ,  are  x  =  --  r,  and  y  =  -  ,  —  .      Upon  these  values 
a  —  a  a  —  a 

we  may  observe  that  for  a  =  a',  and  b  and  b'  unequal,  the  values  of  x  and 
y  become  GO.  This  indicates  that  the  Hues  do  not  intersect,  and  hence  that 
they  are  parallel.  Therefore  a  =  a'  is  Ike  condition  of  parallelism  of  two 
straight  lines.  This  may  also  be  seen  directly  from  the  meaning  of  a  and  a'. 
As  these  quantities  are  the  tangents  of  the  angles  which  the  lines  make 
with  the  axis  of  abscissas,  it  follows  that  when  they  are  equal  the  lines  make 
equal  angles  with  this  axis,  and  are,  therefore,  parallel.  2nd.  If  b  =  b',  and 
a  and  a'  are  unequal,  we  have  x  =  0,  and  y  =  b  =  b'.  This  is  also  evident 
from  the  meaning  of  b  and  b'.  Both  lines  cut  the  axis  of  ordinates  at  the 
same  point.  3rd.  If  a  =  a  and  b  =  b',  x  =  ft,  and  y  =  ft,  and  the  lines 
coincide.  4th.  If  a  =  0,  and  6  =  0,  the  first  equation  becomes  y  =  0,x  -j-  0, 

or  the  equation  of  the  axis  of  abscissas,  and  x  =  ---  -,  the  point  of  inter 
section  of  y  =  ax  -f-  V  with  this  axis,  as  it  ought.  Sen.  2,  ABT.  33. 

Sen.  2.  —  Any  two  simple  equations  between  two  variables  being  given,  if 
the  lines  they  represent  are  constructed,  and  the  co-ordinates  of  the  points 
of  intersection  measured,  we  have  a  graphic  solution  of  the  equation. 

Ex.  1.  Where  is  the  point  of  intersection  of  the  lines  %x  —  \y  =  1, 

and  y  =  —  2x  +  4?  Ans.,  (2,  0). 

Ex.  2.  What  are  the  co-ordinates  of  the  vertices  of  the  triangle  the 
equations  of  whose  sides  are  y  —  2x  -j-  3  =  0,  —  —  --  -f  4  =  ^y,  and 


Ex.  3.  "Where  does  a  perpendicular  from  (  —  3,  8),  to  the  line  y  = 
—  5,  intersect  the  latter?  Ans.,  At  (1-J-, 


OF   PLANE   ANGLES   AND   THE   INTERSECTION   OF   LINES.  27 

Ex.  4.    Where  does    a   perpendicular    from  the    origin    intersect 


and 


Ex.  5.  Given  y  =  ^x  —  3,  y  =  —  ix  —  8,  and  y  =  —  $x  -f  10, 
as  the  equations  of  the  sides  of  a  triangle,  required  to  find  where  a 
perpendicular  from  the  angle  included  between  the  first  two  sides, 
intersects  the  third  side. 

Result,  At  the  point  5.5,  6.  34,  nearly. 

Ex.   6.    Find-  the   intersections    of    the    loci 
whose  equations  are  7(y  —  x)  =  5  —  2 
y*  -f  x*  +  9  =  16  —  6y,  and  construct 
the  figure. 

Results,  At  the  points  (.374  +,  -981  +), 
and  (—3.888  +,  —2.063  +).  The  figure 
is  that  given  in  the  margin. 

Ex.  7.  Eind  the  intersections  of  yz  = 
10#,  and  xs  -f  \f  =  144,  and  construct 
the  loci,  thus  verifying  the  solution. 


Ex.  8.  Eind  the  intersections  of  25y2+  16x2  =  1600,  and 
+  576  =  0.       Results,  At  (9.12,  3.3),  and  (—9.12,  —3.3),  nearly. 

Ex.  9.  Find  the  intersections  of  x*  —  Sx  -f  i/2  -f  Gy  —  0,  and  y  = 
\x  -f  1.     Also  of  the  first  with  3y  =  4#.     Also  with  y  —  3  —  x. 
Results. — 1st.  Imaginary  results.      No  intersections.     2nd.  A  com 
mon  point  at  the  origin.     3rd.  Two  points  of  intersection. 

Sen.  3. — The  construction  of  loci  represented  by  equations  affords  beau 
tiful  illustrations  of  principles  in  the  theory  of  equations,  concerning  the 
number  and  character  of  the  roots  of  an  equation. 

Ex.  10.    Find    the   intersections  of 
4#2  =  100,  by  the  following  :  1st,  y»  -f 
#»  =  9  ;    2d,    y2  +  2y  -f-  x*  =  8  ;  3d, 

i/3  +  4?/+a;*  =  5;  4th,  y*-f  10y+tf'=     5^ l/r  A'*  '\i TB^C 

—16  ;  and  5th,  y2  +  12y  -f  #2  =  —27. 


Results.— 1st,  At  (2.44, 1.7);  (—2.44, 1.7); 
(2.44,  —1.7);  (—2.44,  —1.7)  which  affords 
an  example  of  4  real  roots.  2nd,  At  (0.2)  ; 
and  (±  2.9,  — |^),  which  affords  an  example 
of  what  seems  to  be  but  three  roots  when  there 
should  be  four.  This  is  explained  by  the 
two  values  of  x  for  y  =  2,  becoming  -f  0  and 


28 


THE   CARTESIAN  METHOD    OF   CO-OBDINATES. 


— 0,  or  practically,  though  not  theoretically,  one.  3rd,  Gives  two  real 
and  two  imaginary  points,  illustrating  that  imaginary  roots  enter  in 
pairs.  4th,  Gives  two  equal  real  roots,  both  0,  and  two  imaginary, 
showing  a  point  of  contact.  5th,  Four  imaginary  roots,  showing  no 
common  point,  the  additional  imaginary  roots  again  entering  in  a 
pair. 


Ex.  11.  Find  the  intersections  of  2y2  —  ±xy  +  2.r2 —  3?/  — 2or —  8  = 
0,  by  4?/2-f  4#2— 11  =  0.  Also  by  i/2  +  2?/-f#2 — 607  +  6=0.  Also 
by  2/9+6?/-f  #* — 4^  +  9=0. 

Results. — By  the  first  in  4  points.  By 
the  second  in  2  points.  By  the  third  not 
at  all.  The  figure  is  seen  in  Fig.  29,  in 
which  a  a  a  is  the  1st  locus,  and  111, 
222,  and  333,  the  others,  in  order. 


A 


41.  I*rob*  To  find  the  perpendicular 
distance  from  a  given  point  to  a  given  line. 

METHOD  OF  SOLUTION. — First,  find  the  equation 
of  a  line  passing  through  the  given  point,  and 
perpendicular  to  the  given  line  (32,  39).  Second,  FIG  29.. 

find  the  point  in  which  this  perpendicular  inter 
sects  the  given  line  (40).    The  problem  then  consists  in  finding  the  distance  between 
two  points  (29. ) 

COB. —  To  find  the  distance  between  two  parallels,  write  the  equa 
tion  of  a. line  perpendicular  to  the  parallels  (39),  and  find  its  intersec 
tions  with  the  parallels.  The  problem  is  then  the  same  as  (29). 

Ex.  1.  Find  the  distances  of  the  following  points  from  each  of  the 
lines  y  =  2*  —  3,  and  ^x  —y  =  — 1,  viz.,  3,  2  ;  —4,  — 1;  0,  — G  ; 
0,0. 

SOLUTION. — To  find  the  distance  from  — 4,  — 1,  to  y  =  2x  — 3,  we  have  for  the 
equation  of  a  line  passing  through  this  point  and  perpendicular  to  this  line 
y  4-  1  ==  —  4(05  +  4),  or  y  =  —  hx  —  3.  The  intersection  is  at  0,  —3.  The 
distance  between  — 4,  —1  and  0,  —3  is  D  =  \/16  -}-  4,  or  ^/20T 

Ex.  2.  Find  the  sides,  the  angles,  and  the  perpendicular  distances 
from  the  angles  to  the  opposite  sides  in  the  triangle  the  equations  of 
whose  sidee  are  36?/  —  4#  =  45,  3y  -f  3  =  — x,  and  y  —  Ja?  —  3. 

The  sides  are  12.79,  7.44,  and  6.79. 
Result*.—  1   The  angles  are  24°46',  127°53',  and  27°21'. 
The  perpendiculars  are  6.23,  3.33,  and  5.36. 


^ 

H 


OF   THE   CONIC   SECTIONS. 


Ex.  3.  The  vertices  of  a  triangle  are  at  2,  8  ;  — 6,  1  ;  and  0, 
required  the  equations  of  the  sides,  of  the  lines  drawn  from  the  vertices 
to  the  middle  of  the  opposite  sides,  and  of  the  lines  drawn  bisectin^ 
the  angles  and  terminating  in  the  opposite  sides. 


SECTION  VL 

Of  the  Conic  Sections, 

42.    RoscovicJi's  Definition  of  a  Conic  Section. — 

A  Conic  Section  is  a  curve,  the  distance  of  any  point  in  which  from  a 
given  point,  is  to  its  distance  from  a  given  straight  line,  in  a  given 
ratio.  If  the  distance  to  the  point  is  equal  to  the  distance  to  the  line, 
the  locus  is  a  Parabola  ;  if  less,  an  Ellipse  ;  if  greater,  an 
Hyperbola.  If  the  distance  to  the  line  is  infinite,  the  locus  is  a 
Circle  ;  but  if  the  distance  to  the  point  is  infinite,  the  locus  is  a 
Straight  Line. 


To  construct  a   Conic  Section  from  Boscovich's   de- 


43.  I>rob. 

finition. 

SOLUTION. — Let  F,  Fig.  30,  be  the  given  point, 
A  B  the  given  line,  and  m  :  n  the  given  ratio. 
Through  F  draw  C  K  perpendicular,  and  G  H 
parallel  to  AB.  Take  FG  (=  FH)  :  FC  :  : 
m  :  n,  and  draw  CG  and  CH,  producing  them 
indefinitely.  Draw  a  series  of  parallels  to  G  H, 
meeting  the  lines  C  M  and  C  N.  Now  with  the 
half  of  any  one  of  these  lines,  as  L_~T,  for  a  radius, 
and  the  given  point,  F,  as  a  centre,  describe  an 
arc  cutting  the  parallel  taken,  as  at  P.  Then  is 
P  a  point  in  the  curve.  To  prove  that  P  is  a 
point  in  the  curve,  join  P  and  F,  and  draw  PR 

parallel  to  C  K.  By  similar  triangles  we  then  have  FIG.  30. 

LT(=  PFj  :  TC  (  =  PR)  :  :  G  F  :  FC   (by  construction)  :  :  m  ':  n.  .-.  PF  : 
3R  :  :  m  :  n.     In  like  manner  any  required  number  of  points  in  the  curve  may  be 
determined,  so  that  by  connecting  them  the  curve  will  be  completely  drawn.     In 
this  figure,  as  FG  <  FC  the  curve  is  an  ellipse.     Had  FG  been  taken  equal  to 

C,  the  curve  would  have  been  a  Parabola.     And  if  FG  had  been  greater  than 
FC,  the  curve  would  have  been  an  Hyperbola. 

44.  DEFS.— The  fixed  line,  AB,  is  the   Directrix.    The  fixed 
point,  F,  is  the   Focus.     CM   and  CN  are    the   Focal  Tan- 
gents. 


30 


THE   CARTESIAN   METHOD   OF   CO-0  KDINATES. 


The  portion  of  the  perpendicular  to  the  directrix  through  the  focus, 
CK,  intercepted  by  the  curve  is  the  Transverse  or  Major  Aocis, 
as  I  K.  The  centre  of  the  transverse  axis,  O,  is  the  centre  of  the  curve. 

The  perpendicular  to  the  tranverse  axis  passing  through  the  centre, 
and  limited  by  the  curve  (in  the  ellipse),  as  DE,  is  the  Conjligate, 
or  Minor  Axis.  The  double  ordinate  passing  through  the  focus, 
GH,is  the  Latus  Hectum,  Principal  Parameter,  or  the 
parameter  to  the  transverse  axis.  The  extremities  of  the  transverse 
axis,  I  and  K,  are  the  Vertices.  The  distances  from  the  focus  to  the 
vertices  are  the  Focal  Distances.  The  Eccentricity 9  is  the 
distance  from  the  focus  to  the  centre,  divided  by  the  semi-transverse 


axis, 


FO 


Ex.  1.  Construct  a  parabola  whose  parameter  is  12. 


CONSTRUCTION.  — Let  A  B  be  the  directrix.  Draw 
C  K  perpendicular  to  it,  take  F  at  a  distance  6  from 
the  directrix,  and  through  F  draw  G  H  parallel  to 
A  B.  Take  G  F  =  H  F  =  6,  and  draw  CM,  and 
C  N  through  G  and  H .  (The  construction  is  then 
completed  as  in  the  problem  above.)  To  show  that 
any  point  thus  found,  as  P,  is  a  point  in  a  parabola 
whose  parameter  is  12,  observe  that  LT  (=PF): 
CT  (=  PR)  :: ;GF  :CF.  ButGF  =  CF  =  6, 
by  construction.  .-.  PF  =  PR.  AlsoGH, 
the  parameter,  =  2G  F  =  12. 

QUERIES. — Can  the  parabola  ever  return  into  itself 
so  as  to  inclose  a  space  ?  Why  ?  Can  portions  of  this 
curve  lie  on  both  sides  of  the  directrix  ?  Why  ? 


FIG.  31. 


43.  COR.  1. — Jfp  —the  focal  ordinate,  i.  e.,  one  half  the  Latus  Rectum, 
the  vertex  is  at  ip/rom  the  directrix,  and  the  distance,  IF  =  £p,  or  \ 
the  parameter,  G  H . 

Ex.  2.  Construct  an  ellipse  in  which  the  fixed  ratio  shall  be  §,  and 
the  distance  from  the  focus  to  the  directrix  6.  Also  with  the  ratio  £ 
and  the  distance  from  the  focus  to  the  directrix  5. 

QUERIES.— With  the  same  focus  and  directrix  how  does  varying  the  ratio  affect 
the  form  of  the  curve  ?  With  the  same  ratio  and  directrix  how  does  varying  the 
position  of  the  focus  affect  the  form  of  the  curve  ?  How  does  it  appear  from  the 
first  query  that  when  the  ratio  is  0,  the  locus  is  a  point  ? 

Ex.  3.  Construct  an  ellipse  whose  latus  rectum  shall  be  6,  and  the 
fixed  ratio  . 


OF  THE  CONIC  SECTIONS.  31 

SUG.— In  Fig.  30,  if  Q  F  =  3  and  the  characteristic  ratio  of  the  curve  is  ^,  what 
isCF? 

46.  COR.  2 — From  Fig.  30,  by  principles  of  construction  it  appears  that 
the  tangents  at  the  vertices,  viz.,  IS,  and  KM,  are  equal,  respectively,  to 
the  focal  distances  I  F,  and  F  K-  It  also  appears  that  the  distance  from 
the  focus  to  the  extremity  of  the  conjugate  axis  in  an  ellipse,  F  D,  equals 
the  semi-transverse  axis ;  for  F  D  =  Q.O  =  K I  S  +  M  K )  =  i  I  K- 

Ex.  4.  Construct  an  ellipse  whose  transverse  axis  shall  be  12,  and 
conjugate  10  ;  i.  e.,  having  given  the  axes,  to  construct  the  ellipse. 

Ex.  5.  Construct  an  ellipse  whose  transverse  axis  shall  be  10,  and 
distance  between  the  foci  8. 

Ex.  6.  Having  given  the  curve  and  the  transverse  axis,  to  find  the 
foci  and  directrix  of  an  ellipse. 

SOLUTION.—  Let  ACBC',  Fig  32,  be  the  curve,  and 
A  B  its  transverse  axis.  Bisect  the  transverse  axis  with 
a  perpendicular,  and  the  portion  of  this  perpendicular 
intercepted  by  the  curve  will  be  the  conjugate  axis. 
From  either  vertex  of  the  conjugate  axis  as  a  centre,  with 
a  radius  equal  to  the  semi-transverse  axis,  describe  arcs 
cutting  the  transverse  axis  ;  these  points  will  be  the  foci 
(4-0).  As  there  are  two  intersections,  there  are  two  foci. 

At  each  extremity  of  the  transverse  axis  erect  perpendiculars  and  make  them 
severally  equal  to  the  adjacent  focal  distances,  thus  obtaining  two  points  in  the 
focal  tangent  (46).  Draw  the  focal  tangent,  and  where  it  intersects  the  transverse 
axis  produced,  erect  a  perpendicular  to  this  axis,  and  this  perpendicular  will  be  the 
directrix. 

QUERIES. — How  does  it  appear  from  the  definition  of  the  ellipse,  that  the  curve 
can  not  lie  on  both  sides  of  the  directrix  ?  How  does  it  appear  that  the  curve  cuts 
the  axis  beyond  the  focus  ? 

Ex.  7.  Letting  A  represent  the  semi-transverse  axis,  B  the  semi- 
conjugate,  2c  the  distance  between  the  foci,  and  e  the  eccentricity^ 
show  that 

_  c    _ 
~~A~ 

B* 
and  hence  that  1  —  e2  =  — -  •   How  does  it  appear  from  this  that  in 

the  case  of  the  ellipse  e  <^  1  ? 

Ex.  8.  Construct  an  ellipse  whose  transverse  axis  is  12,  and  eccen- 
centricity  f . 

SUG.     First  find  the  value  of  J5,  which  is  4£,  nearly. 


32 


THE  CARTESIAN  METHOD  OF  CO-OEDINATES. 


Ex.  9.     Construct  an  hyperbola. 

SOLUTION. — Let  AB  be  the  directrix, 
F  the  focus,  and  m  :  n  the  ratio,  iu 
which  m  >  n.  Through  F  draw  F  K 
perpendicular  to  the  directrix  and  GH 
parallel,  producing  both  indefinitely. 
Take  FG  (=  FH)  :  CF  :  :  m  :  n. 
Through  C  and  G  draw  MM',  and 
through  H  and  C,  NN',  the  focal  tan 
gents.  (The  process  is  exactly  analogous 
throughout,  to  that  pursued  in  construct 
ing  the  ellipse,  and  hence  need  not  be 
detailed.  The  student  can  supply  it.  It 
should  be  noticed,  however,  that  the 
distance  from  the  focus  to  any  point  in 
the  curve,  being  greater  than  the  distance 
FIG.  33.  from  the  same  point  to  the  directrix,  there 

may  be  (are)  points  in  the  curve  on  the  opposite  side  of  the  directrix  from  the  focus. 
These  points  are  determined  in  the  same  manner  as  the  others.  Thus  the  point 
Pvn  is  found  by  taking  T'N'  as  a  radius,  and  from  F  as  a  centre  drawing  an  arc 
cutting  T'N'  in  Pvn.  In  like  manner  other  points  in  this  branch  are  located.) 

The  demonstration  is  as  follows  :  To  prove  that  any  point,  as  P ,  is  in  the  curve, 
we  have  to  prove  that  PF  :  PR  : :  m  :  n  ;  t.  e.,  the  distance  from  any  point  in 
the  curve  to  the  focus,  is  to  the  distance  of  the  same  point  from  the  directrix,  in  a 
constant  ratio  (m  :  n),  which  ratio  is  greater  than  1,  in  the  hyperbola.  To  prove 
that  the  construction  gives  this  proportion,  join  P  and  F>  and  draw  PR  parallel 
to  TC,  Now  since  PF  =  LT,  and  PR  =  TC,  and  by  reason  of  similar  trian 
gles,  we  have  PF  :  PR  : :  LT  :  TC  :  :  GF  :  FC  : :  m  :  n.  In  a  similar  manner 
any  point  on  the  other  side  of  the  directrix,  found  by  the  method  described,  as 
PVTI,  is  shown  to  be  in  the  curve.  Thus  PVI1F  =  N'T'  by  construction,  PV»R'  = 
T'C,  and  the  triangles  CFG  and  CT'N'  are  similar  ;  hence  PVIIF  :  PV1:R'  : : 
T'N'  :  T'C  : :  G  F  :  FC  :  m  :  n.  Q.  E.  D. 

47.  DEF'S. — The  Axis  of  the  Hyperbola  is  an  infinite  line 
drawn  through  the  focus  and  perpendicular  to  the  directrix,  as  TT^ 
Fig.  33. 

TJie  Transverse  Axis  of  the  Hyperbola  is  that  por 
tion  of  the  axis  of  the  curve  included  between  the  vertices,  as  K I , 
Fig.  33. 

TJie  Focal  Distances  are  the  distances  from  the  focus  to  the 
vertices,  as  Fl,  and  F  K,  Fig.  33. 

The  Conjugate  Axis  of  the  Hyperbola  is  a  perpendic 
ular  to  the  transverse  axis  at  its  centre,  and  is  limited  by  an  are, 
drawn  from  the  vertex  as  a  center,  with  a  radius  equal  to  the  distance 
from  the  focus  to  the  centre.  Thus,  in  Fig.  33,  D  E  represents  the 


OF  THE  CONIC  SECTIONS. 


33 


conjugate  axis,  the  extremities  D  and  E  being  determined  by  making 
the  distances  D  I  and  E I  each  equal  to  OF.  This  definition  is  a 
convention  adopted  for  the  purpose  of  rendering  more  close  the  ana 
logy  between  this  curve  and  the  ellipse. 

A  Conjugate  Hyperbola  is  an  hyperbola  having  the  conju 
gate  axis  of  a  given  hyperbola  for  its  transverse  axis,  and  the  trans 
verse  axis  of  the  given  curve  for  its  conjugate ;  see  Ex.  10,  Fig.  34. 
Either  of  two  hyperbolas  thus  related  is  conjugate  to  the  other.  They 
are  sometimes  distinguished  as  the  X  hyperbola  and  the  Y  hyper 
bola,  each  taking  the  name  of  the  co-ordinate  axis  upon  which  its 
transverse  axis  lies. 

An  IEq^lilateral  Hyperbola  is  one  which  has  its  conjugate 
axis  equal  to  its  transverse. 


Ex.  10. 
8  and  6. 


To  construct  a  pair  of  conjugate  hyperbolas  whose  axes  are 


SUGS. — Draw  two  indefinite 
straight  lines  at  right  angles  to 
each  other,  and  take  Ol  = 
OK  =4,  and  OD=OE  =  3. 
Having  constructed  the  branches 
on  the  axis  Kl,  Fig.  34,  as  in 
Ex.  9,  take  O  F'  =  OF  (which 
=  ID),  and  F' is  the  focus  of 
the  conjugate  or  Y  hyperbola. 
Taking  DS'  =  D  F'  and  E  L  == 
EF',  and  through  S'  and  L' 
drawing  a  right  line,  it  is  one 
of  the  focal  tangents.  Having 
found  the  focal  tangents  the 
construction  proceeds  as  before. 


FIG.  34. 


Ex  11.  Construct  an  hyperbola  whose  transverse  axis  is  G,  and 
less  focal  distance  2.  Eind  also  the  conjugate  axis,  focus,  and  direc 
trix  of  the  conjugate  hyperbola. 

Ex.  12.  Letting  e  represent  the  eccentricity  of  an  hyperbola,  c  the 
distance  from  the  centre  to  the  focus,  A  the  semi-transverse  axis,  and 
B  the  semi-conjugate,  show  that 


and  hence  that  1  —  e9  = 

'  A 2 

greater  than  1  in  the  hyperbola  ? 

Ex.  13.     What  is  the  eccentricity  of  an  hyperbola  whose  axes  are 


34  THE  CARTESIAN  METHOD  OF  CO-ORDINATES. 

10  and  6  ?    What  is  the  eccentricity  of  an  hyperbola  whose  transverse 
axis  is  12,  and  less  focal  distance  3  ? 

Ex.  14.  The  eccentricity  being  1^  and  the  conjugate  axis  4,  what 
is  the  transverse  axis  ?  What  the  focal  distances  ?  What  the  charac 
teristic  ratio  (42)  ?  Transverse  Axis,  3.577  +. 


48.     I*rop. 


Boxeovich't  ratio  and  the  eccentricity  are 

DEM.— 1st.  Let  AB,  Fig.  35,  be  the  directrix 
of  an  ellipse,  F  the  focus,  CL  the  focal  tangent, 
and  O  the  centre.  Draw  GK  parallel  to  CL- 
Then  since  LO  =  GO,  and  GF  =  IG  =  LK 
(46),  LO  -  LK  -  KO  =  GO  -  GF  =  FO. 


By  defi- 


FIG. 


Bimilar  triangles,  = 


K  O 

Therefore,  =  the  eccentricity  (44). 

GO 

nition  =  =  Boscovich's  ratio.     Now,  by 

CG        CG 


Q.  E.  D. 

2nd.     In  the  Hyperbola  the  demonstration  is 
essentially  the  same.     Thus,  in  Fig.  3G,  LO  = 
—    IG         FE  —  FG 


QF    —     G     —   the   characteristic   ratio    (Bos- 


Q.  E.  D. 


~~2~~  2 

LO  +  LK  =  KO  =  GO  +   FG   =   FO, 

|^  (*± 

and  — —    =  the    eccentricity.      By    definition 
GO 

QF   __  IG 
CG        CG 

covich's     ratio).      Now,    by    similar    triangles, 
IG  __  KO 
CG        GO 

FIG.   36.  3rd.     In  the  Parabola  we  may  call  the  eccen 

tricity  1   from  analogy  ;  or,  better,  we  may  conceive  the  parabola  to  be  an  ellipse 
with  the  centre,  O,  removed  to  an  infinite  distance  from  the  vertex,  G,  Fig.  35, 

whence  the  fraction  — —  =  1.*     Q.  E.  D. 

'49.     COR. The  student  should  fix  in  memory  the  following  relations, 

as  they  are  fundamental,  and  of  frequent  use  in  the  reduction  offormulce. 

Letting  A  =  the  semi-transverse  axis,  B  =  the  semi-conjugate  axis, 
e  =  the  eccentricity,  and  p  =  the  semi-lotus  rectum,  we  have  the  fol 
lowing 

*  If  the  student  has  difficulty  in  understanding  this  statement,  let  him  consider  that,  O  being 
removed  to  infinity,  the  finite  distance,  G  F,  by  which  GO  appear,  to  be  greater  than  FO,  is  of  no 
appreciable  value  as  compared  with  the  terms  of  the  ratio,  which  are  both  infinite. 


OF   THE    CONIC    SECTIONS. 


35 


FUNDAMENTAL  RELATIONS. 

From  the  definition  (42)  and  (48),  we  see  that,  The  distance  from 
any  point  in  the  curve  to  the  focus  -f-  e  =  the  distance  from  the  same 
point  to  the  directrix.  Also,  The  distance  of  any  point  in  the  curve 
from  the  directrix  X  e  =  the  distance  of  the  same  point  from  the  focus. 


Diftntices. 

JHUipse. 

HypftrlMtlit  . 

I'tl  l'(ll>oltl 

1. 

(  Focus  to  extremity  of  CONJ.-AXIS.  .   = 
(  VERTEX         "                   "         = 

A 

Ae 

GO 
GO 

2 

Focus  to  CENTRE  .                 .              .  .  •  —  - 

Ae 

Ae 

GO 

3. 

FOCAL  DISTANCES  ....  — 

A(l  =p  e) 

A(e  =F  1) 

$P 

4" 

A(\  +  e) 

A(e  =F  1) 

ln 

e 

e 

*tr 

it 

A(l  —  e*) 

A(e-  —  li 

e 

e 

P 

Q 

CENTRE  to  DIRECTRIX                   .           — 

A 

A 

•       e 

e 

7. 

1  —  &               = 

& 
A* 

B* 
A* 

0 

8 

SEMI-LATUS  RECTUM,  p    = 

JB* 

& 

D 

A 

A 

DEM. — For  the  ellipse  see   Fig.  30,  for   the  hyperbola, 
Fig.  33,  and  for  the  parabola,  Fig.  37. 

(1.)  For  Ellipse  see  (46).— For  Hyperbola,   by  definition 

of  eccentricity  — —  =e.      .'.    FO  =  Ae  =  the    distance 
A 

from  the  vertex  to  the  extremity  of  the  conjugate  axis,  by 
the  definition  of  the  latter  (47)-      For  Parabola,   consider 
the  curve  as  an  ellipse  with  its  centre  removed  to  infinity. 
cro 

FO  ===  Ae. 


(2.)  For  Mipse,     -      = 
A 


by  definition. 


FIG.  37. 


For  Hyperbola  and  Parabola,  see  above. 

(3.)    For  Ellipse,    I  F  =  I  O  —  FO  =  A  —  Ae  =  A(l  —  e).       FK  = 
-f-  FO  —  A  -j-  Ae  =  A(l-^-e).     For  Hyperbola,    I  F  =  FO  —  I  O  —  Ae 
A(e  —  l\      FK  =  FO  +  OK  =  Ae-{-  A  =  A(e+  1).     For  Parabola  see 

I  tr       A(i e} 

(4.)  For  Ellipse,  since  I  is  a  point  in  the  curve    1C  = = . 


OK 

A  = 


KC 


KF 


For    Hyperbola,    for     same    reason     I C    = 
1) 


Also, 
IF 


(5.)    For  Ellipse,    FC  =  \F  -\-  \G  =  A  —  Ae 


For  Parabola,  see 

4U  — «) 


36  THE   CAETESIAN   METHOD   OF   CO-ORDINATES. 

For  Hyperbola,    FC  =  I  F  -f  I  O  =  .4e  _  ^  4.  — --  =  A(e"  ~1}    For  Par- 

e  e 

abola,  Fp  =  G  F  =  p,  by  definition. 

(6.)  For  Ellipse,  D  being  a  point  in  the  curve  whose  distance  from  the  direc 
trix  is  OC,  we  have  OC  = =  — .  For  hyperbola,  OC  =  O  I  —  C  I  = 

66  • 

— =  -^--     (The  distance  in  the  ellipse  may  be  obtained  in  the  same 

way.)     For  Parabola,  same  conception  as  in  (1)  above. 

(7.)  For  Ellipse  and  Hyperbola  see  Ex's.  7  and  12.  For  Parabola,  I  —  e2  =  1  —  1 
=  0. 

(8. )  For  Ellipse,  G  F  =  p  ==  F  C  X  e  =  A  (1  —  e*)  =  ^-.      For   Hyperbola,   G  F 

A. 
B~ 
=  p=  FCXe  =  A(e'2  —  1)  =  — .     For  Parabola,  G  F  =p  by  definition. 

*^»  I?vob»  To  pass  a  conic  section  through  three  given  points,  so 
that  it  shall  have  a  given  focus  ;  and  to  determine  its  elements  ;  i.  e.,  the 
axes,  foci,  directrix,  eccentricity,  etc.,  if  an  ellipse  or  hyperbola,  or 
the  lat-us  rectum  if  a  parabola. 


SOLUTION.  —  Let  M,  N,  and  O,  Fig.  38,  be  the 
given  points,  and  F  the  given  focus.  Connect  the 
points  with  the  focus,  and  draw  ON,  and  N  M, 
and  produce  them  towards  the  probable  position 
of  the  directrix,  as  to  L  and  K.  Now,  take  a 
point  R,  on  O  L,  such  that  OF  :  N  F  :  :  OR  : 
NR,*  and  R  is  a  point  in  the  directrix.  In  like 
manner,  take  N  F  :  M  F  :  :  N  S  :  M  S,  and  S  is 
another  point  in  the  directrix.  Hence  the  direc 
trix  can  be  drawn. 

To  prove  that  a  line  drawn  through  R,  and  S, 
as  A  B,  is  the  directrix,  we  have  to  show  that 


being  perpendicular  to  A  B.     Now  OP:  NQ  ::OR  :  NR.     But  by  construc 
tion  OR  :  NR  ::  OF  :  N  F.  .-.   OP  :  OF  ::  NQ  :  N  F,    or   ~   =    MZ.. 

In  like  manner  NQ  :  MT  ::NS:MS::NF:MF.    .-.    -^    =    MF 

NQ          MT' 

Q.  E.  D. 

To  make  the  numerical  computations  requires  much  more  labor  than  to  effect 
the  geometrical  solution.  We  may  proceed  as  follows  :  Having  the  distances  O  N  , 
N  M.  OF,  N  F,  and  M  F  given  in  numbers,  compute  the  numerical  values  of 
N  R  and  M  S  from  the  proportions  used  in  the  construction.  The  sides  of  the 
t.iaugles  O  F  N  and  N  F  M  being  known,  their  angles  can  be  found  by  trigo- 


*  This  construction  i.?  effected  thus:  taking  the  proportion  by  division,  (OF  —  N  F),  or  OG  :  OF 
:  :(OR  —  NR),  or  ON:  OR.  From  this  proportion  O  R  can  be  constructed,  as  the  other  terms  are 
known. 


OF  THE  CONIC  SECTIONS. 


37 


nometry;  whence  we  get  the  angle  R  N  S,  as  it  equals  180°—  (O  N  F  -f-  FN  M). 
Then,  in  the  triangle  RNS,  we  shall  have  two  sides  and  the  included  angle; 
whence  the  angle  N  RS  can  be  found,  and  from  it  N  RQ  becomes  known.  Now, 
in  the  right  angled  triangle  N  RQ,  we  know  the  hypothenuse  and  one  acute  angle, 
and  can  find  NQ.  Again,  letting  fall  the  perpendicular  N  E,  forming  the  trian 
gle  F  N  E,  we  can  compute  E  F,  since  F  N  is  known  and  the  angle  F  N  E  = 
FNR-fRNQ  —  90°.  FC  is  therefore  known,  being  equal  to  EF  -f-  NQ. 

N  F 

As  —  -  is  now  determined,  the  ratio  e,  or,  what  is  the  same  thing,  the  eccen- 


the 


tricity,  is  known.     Taking  a  point,  as  H,  upon  FC,  such  that 


H  C          N  Q 

vertex  is  determined.  In  a  similar  manner  the  other  vertex  of  an  ellipse  or  hyper 
bola  Can  be  found.  Letting  p  be  half  the  lotus  rectum,  F  U  ,  it  can  be  found  from 
FN  p 

NQ  ~~   FC' 

Ex.  1.  Construct  a  conic  section  passing  through  the  points  O,  M, 
and  N,  and  having  F  for  a  focus,  knowing  that  O  F  =  6£,  N  F  =  3£, 
M  F  =  2-i  ,  O  N  =  v/18,  N  M  =  v/10.  Let  the  geometrical  con 
struction  be  given,  and  also  the  numerical  solution. 

The  locus  is  a  parabola  whose  latus  rectum  is  9. 

Ex.  2.  Construct  and  compute  as  above,  when  O  F  =  2.08,  N  F 
1.08,  M  F  =  .46,  ON  =  1.12,  and  N  M  =  .87. 

Ex.  3.  Construct  and  compute  as  above,  when  O  F  =  10,  N  F  = 
C,  M  F  =  3,  ON  =  6,  and  N  M  =  4. 


51.     IProb.     To  produce  the  general  equation  of  a  Conic  Section 
referred  to  rectangular  axes. 


SOLUTION.—  Let  M  N,  Fig.  39,  be  an  arc  of 
any  conic  section,  F  the  focus,  C  B  the  direc 
trix,  ZZ'  the  axis  of  the  curve,  and  AX  and 
AY  the  axes  of  reference.  Let  P  be  any  point 
in  the  curve,  and  its  ordinate  P  D  :  also  draw 
the  ordinate  of  the  focus,  FK.  Draw  from 
the  origin  AG  perpendicular  to  CB.  Draw 
DH  parallel,  and  P L  perpendicular  to  CB. 
X  Join  P  and  F,  and  draw  PI  parallel  to  AX. 

Let  AG,  the  distance  from  the  origin  to  the 
directrix,  be  represented  by  d  ;  the  co-ordinates 


~B 


FIG.  39. 

of  the  focus,  A  K  and  F  K,  by  m  and  n  respectively  ;  the  ratio  mentioned  in  the 

P  F 

definition  (42} ,   p~^.»  by  e  ;  the  angle  which  the  axis  of  the  curve  makes  with  the 

axis   of  abscissas,    ZSX^GAX^LDP,    by  a;    and  the  general  co-ordi 
nates,  A  D  and  PD,  by  x  and  y. 


38 


THE   CAETESIAN   METHOD   OF   CO-ORDINATES. 


Now,  J=»R  =e*  -  PE'.      But  PF-  =  PI*  -f   Fl*  =  (m  -  3)*  +  (n  —  T/)*. 
Again,    PE2  ==  (PL  -f-  AH  —  AG)2  =  (PD  sin  a  -f  AD  cos  a  —  d)2  = 

(y  sin  a  -f-  a  cos  a  —  d)2.     Whence,  substituting  we  have 

(£?.  4),     (m  —  x)'2  -f  (n  —  ?/)2  =  e'2(?/  sin  «  -f  x  cos  a  —  d)2. 

#£.     COR.  1.  —  The  equation  of  the  ellipse  and  hyperbola  referred  to 
their  axes  is 


in,  which  A  is  #ie  semi-transverse  axis,  and  e  #ie  eccentricity,  or  the  char 
acteristic  ratio  (42). 


DEM. — Let  the  curve  in  Fig.  39  be  conceived 
to  change  position  so  as  to  assume  that  in  Fig. 
40  or  41,  the  axis  of  the  curve  coinciding  with 
the  axis  of  abscissas,  the  origin  at  the  centre, 
A,  F  the  focus  and  CB  the  directrix.  Aa 
the  axis  of  the  curve  now  coincides  with 
the  axis  of  abscissas,  a  =  0.  As  the 
focus  falls  upon  the  axis  of  abscissas,  n  —  0. 
By  consulting  (40)  it  will  be  seen  that 
m  =  ^=  A  F  =  =p  Ae,  and  d  =  q=  AG  =  =F 

A 

— ^.  "Whence,  substituting  these  values  in 

Eq.   A,   we  have  (  =p  Ae  —  x)'2  -f-  y*  = 
A)2.       Expanding 

and  reducing,  we  have 

(Eq.    B),  y*  -f  (1  —  e2)o58  =  ^2(1  —  e2). 

Q.  E.  D. 

QUERY. — How  is  it  that  the  same  equa 
tion  is  made  to  represent  two  curves  so 
different  from  each  other? 


FIG.  41. 


53.  COR.  2.  —  The  equation  of  the  ellipse  referred  to  its  axes,  and  in 
terms  of  its  semi-axes  is 


in  which  A  and  B  are  the  semi-axis. 


*  The  upper  sign  applies  to  the  ellipse,  and  the  lower  to  the  hyperbola. 

The  same  final  result  may  be  obtained  in  the  case  of  the  hyperbola  by  conceiving  the  axis  ZZ' 
of  Fig.  39  to  have  revolved  to  the  left  and  the  branch  to  fall  on  the  left  of  the  origin.  In  this 
case  (X.  =  180°,  cos  Oi  =  —  1,  and  *  is  negative. 


OF   THE    CONIC    SECTIONS. 


39 


DEM.—  This  equation  is  obtained  at  once  from  Eq.  B  by  substituting  for  1  —  e* 


T)> 


> 

its  value,  —  (49),  and  clearing  of  fractions. 
A.~ 

54.  COR.  3.—  The  equation  of  the  hyperbola  referred  to  its  axes,  and 
in  the  terms  of  its  semi-axes,  is 


in  which  A  and  B  are  the  semi-aj;es. 
DEM.—  Substituting  in  Eq.  B,  -  ^  for  1  —  e2  (49),  we  nave,  after  clearing  of 

fractions,  A-y*  —  BW  =  —  A*B*. 
SCH.—  These  two  equations,  viz., 

A*ys  4-  B*x*  =  A*B*,  for  the  ellipse,  and 
A»yi  _  B*  x*  =  —  A*B2,  for  the  Hyperbola, 

are  the  most  important  forms  of  the  equations  of  these  loci,  and  are  always 
meant  when  their  equations  are  spoken  of  without  specification.  They  may 
be  called  the  Common  Forms. 

55.  COR.  4.—  The  equation  of  the  ellipse  referred  to  its  transverse  axis 
and  a  tangent  at  the  left  hand  vertex,  is 


and  of  the  hyperbola, 


DEM.—  In  either  of  the  annexed  figures, 
let  AX  and  AY  be  the  axes,  F  the 
focus,  and  C  B  the  directrix.  Now  a  —  0  ; 


m 


Ae    I   A 
-j-  AG  = — —  in  the  hyperbola,  hence 

in  general  d  =  Ae  ~^  A  (49).  Substitut 
ing  these  values  in  Eq.  A  (51\  and  reduc 
ing,  we  have 

3/2  -f-  (1  —  e2)x2  —  2A(l  —  e*)x  =  0, 
which  is  applicable  to  either  locus.     For       S     Y' 

1  —  e2  substituting  —  for  the  ellipse,  and 
"-  for  the  hyperbola,  we  have 

2/2  =  —  (2 Ax  —  a;2),  and 


FIG. 


2/2=-IF(2«x- 


Q. 


40  THE   CAliTESIAN    METHOD   OF   CO-OKDINATES. 

50*  COR.  5. — The  equation  of  the  circle  referred  to  a  pair  of  diameters 
at  right  angles  to  each  other  is 

yi  -f  x'2  =  R2,  THE  COMMON  FORM. 

When  the  reference  is  to  a  diameter  and  a  tangent  at  its  left  hand  extremity, 
the  equation  of  the  circle  is 


DEM.  —These  two  forms  are  readily  produced  from 

—  a:2),    by 


-f  £2  9.2 


*~,  and 


making  A  =  B  =  E,  the  radius  of  the  circle,  and 
dividing  out  the  common  factor  R2.  For  the  first 
form  the  origin  is  at  A,  Fig.  43,  and  for  the  second 
form  at  A'. 


#7»  COR.  6. — Making  A  =  B  in  the  equations  of  the  hyperbola,  CORS. 

3rd  and  4th,  there  result  y2  • —  x2  =  — A2,  and  y2  =  — (2 Ax  —  x2),  as 
equations  of  the  Equilateral  Hyperbola  (47). 

58.  COR.   1. — To  obtain  the  equation  of  the  hyperbola  conjugate  to  A2y2 
—  B2x2  =  —  A2B-,  and  referred  to  the  same  axes,  loe  have  bat  to  change 
the  sign  of  the  absolute  term,  and  write  A2y2  —  B2x2  =  A2B2. 


DEM.  —  Conceive  the  curve  Fig.  39,  to  take 
the  position  M  N  ,  Fig.  44,  the  axis  of  the 
curve  and  the  focus  falling  on  the  axis  of 
ordiiiates,  and  the  origin  being  at  the  cen 
tre,  A.  We  then  have  a  =  90°,  m  =  0, 

n  =  AF   =  Be,  and  d  =  AG    =  —  . 

6 

Whence,  substituting  in  Eq.  A,  it  becomes 
a--2  _j_  (Be  —  yY  =  e2  (y  —  -J=(ey—BY  ; 

or,  expanding  and  reducing,  x2  -f  (1  —  e2) 
t/2  =   B*   (1  —   e2). 


But  1  —  e2  = 


— ,  which  substituted,  gives  after  reduction,  A^y*  —  BW  =  A*B*.    Q.  E.  D. 


SO.  COR.  8. — The  equation  of  the  Parabola  referred  to  its  axis  and 
a  tangent  at  the  vertex,  is  y2  =  2px,  in  which  2p  is  the  latus  rectum. 


OF  THE  CONIC   SECTIONS. 


DEM.— Resuming  Eq.  A,  and  conceiving  the 
curve  situated  as  in  Ing.  45,  a  =  0,  n  =  0,  ra 
=  AF  =  ±p,  d  =  —  AG  =  —  ±p,  and  e 
=  1.  Substituting  these  values,  we  have 
(ip  —  a)2  4-  t/2  =  (as  +  ip)2  5  or»  reducing, 
y-  —  2px.  Q.  E.  D.  This  is  the  Common  Equa 
tion  of  the  parabola. 


FIG.  45. 

00.  SCH. — It  will  be  observed  that  the  difference  in  form  between  the 
equation  of  the  ellipse  in  terms  of  the  semi-axes,  and  the  corresponding 
equation  of  the  hyperbola  may  be  considered  as  embraced  in  the  sign  of  B-  ; 
so  that  hereafter,  in  any  case  when  a  property  of  the  ellipse  is  deduced 
from,  the  equation  of  that  locus — as  many  will  be, — the  corresponding 
property  of  the  hyperbola  can  be  discovered  by  simply  substituting  —  B~ 
for  B2  in  the  result ;  or  if  B  only  is  involved,  by  replacing  it  by  B  V  -  -  1. 
Our  subsequent  work  may  often  be  much  abridged  by  this  means.  It  is 
also  to  be  remarked  that,  if  the  formula  expressing  any  property  of  either 
locus  does  not  contain  B,  i.  e.,  does  not  depend  upon  the  conjugate  axis, 
such  property  is  the  same  in  both  loci. 


01.     Prop.     Every  equation  of  the  second  degree,  between  two  vari- 
is  an  equation  of  a  conic  section. 

DEM. — Resuming  Eq.  A,  expanding  and  collecting  terms,  we  have 

(1  —  e2sin9o02/2  —  2e-sinacos  axy  -j-(l  —  e2cos2o:)a;2  -f-(2e2cZ  sin  a  —  Zn)y 

4-(2e?d  cos  a  —  2ra;x  +(m2  4-  w3  —e*d*)=  0.     (Eq.  B. ) 
Representing  these  coefficients  in  order  by  A,  B,  C,  etc. ,  we  have — • 
•Ay*  -f  Bxy  4-  0&  -f  Dy  4-  Ex  -f  F  =  0.     (Eq.  A'. ) 

This  is  the  Complete  Equation  of  the  second  degree  between  two  variables  ;  L  e.  it 
contains  every  variety  of  terms,  with  respect  to  the  variables,  which  such  an  equa 
tion  can  have. 

It  now  remains  to  be  shown  that  these  coefficients  may  have  such  values  (by  the 
locus  being  of  a  proper  species  and  properly  situated)  as  to  cause  the  equation  to 
take  any  and  every  given  form.  Dividing  Eq.  A  by  F,  (any  one  of  the  coefficients 
A,  B,  C,  etc.,  would  do  as  well)  and  distinguishing  the  resulting  coefficients  by 
accents,  we  have  A'y*  -f-  B'xy  -f  C'x*  -\-  D'y  -\-  Ex  -f-  1  =  0.  Now  the  five  coef 
ficients,  A,  B',  C',  D',  E',  depend  upon  the  five  arbitrary  constants,  a,  m,  n,  d,  and  e, 
in  such  a  way  that  such  values  may  be  assigned  to  these  last  quantities,  i.  e. ,  the 
locus  may  be  of  such  species  and  so  situated,  as  to  give  the  quantities  A,  B',  Cf, 
D',  #',  severally  any  and  all  required  values.  Hence  every  equation  of  the  second 
degree  between  two  variables  is  an  equation  of  a  conic  section.  Q.  E.  D. 

[NOTE. — If  the  above  demonstration  seems  abstract,  let  the  student  consider  a 
special  case.  For  example,  let  us  inquire  if  2?/2  —  Sxy  -f  1y  —  5x  -f-  4  =  0,  is  an 


42  THE   CAETESIAN  METHOD   OF  CO-ORDINATES. 

equation  of  a  conic  section  ;  and,  if  it  is,  of  what  species  is  the  locus,  and  how 
situated.  Now,  dividing  this  equation  through  by  4,  we  have  $y2  —  \xy  +  \y  — 
fx  +  1  =  0.  Comparing  this  with  the  equation  Ay2  +  B'xy  +  C"x2  +  D'y-{- 

1  —  e-  sin2  a      _.,       —  2e2  sin  a  cos  a 
E'x  +  1=0,  and  remembering  that  A  =    a     — — ,  ±>  =  —  — — -—-  — , 

1 — e2cos2a          _2e-dsina  —  2n  „, 2e2d  cos  a  —  2m 

write  the^iue  following  equations  : 
1  —  e2  sin2  a 1  . 

'     ''     TO2  _^_  H2   _  e2(p  2    ' 

—  2e2  sin  a  cos  a:  3 

(3).    •L~c—       ^-  =  0,  the  coefficient  of  jc2  in  this  special  example  being  0  ; 


2e2d  sin  a  —  2n, 1 

W-  ™o  i  r,"9    .  g2(£f      2 


2e2d  cos  a:  —  2m 5 

*    m^+w*  — c»d2"~   ~i' 

Now,  from  these  five  equations,  the  values  of  the^ue  quantities  a,  m,  n,  e,  and  d, 
can  be  found.  But  these  being  known,  the  species  (determined  by  e)  and  the 
situation  of  the  locus  are  known.  As  a  similar  course  could  be  pursued  with  any 
particular  equation  of  the  second  degree  between  two  variables,  it  is  evident  that 
every  such  equation  represents  some  conic  section.] 


62.  Prol).  To  determine  the  features  of  an  equation  of  the  second 
degree  between  two  variables,  which  characterize  the  several  species  of  conic 
sections. 

SOLUTION.  —  Comparing  Eq.  A'  with  Eq.  B,  we  see  that  A  =  1  —  e2  sin2  a,  B  = 
—  2e2  sin  a  cos  a,  and  C  =  I  —  e2  cos2o:.     Squaring  the  value  of  B,  and  subtract 
ing  from  this  square  4  times  the  product  of  A  and  C,  we  have 
_#>_  ±AC=  4e»  sin2  a  cos2  a  —4(1  —  e2  sin2  a)  (1  —  e2  cos2  a} 

=  4e»  sin2  a  cos2  a  —  4(1  —  e2  sin2  a  —  e2  cos2  a:  -f  e4  sin^a  cos2  a) 
=  4e  '  sin2  a  cos2  a  —  4  +  4e2(sin2  a  +  cos2  a)  —  4e4  sin2  a  cos2  a 
=  —  4  -f-  4e-,  since  sin2  a  +  cos2  a  =  1. 

...    #2—4^(7  =  4(62  —  1). 

Now,  in  the  parabola,  e  =  1  ;  in  the  ellipse,  e  •<  1  ;  and  in  the  hyperbola,  e  ;>  1. 

Therefore, 

j?2  —  4.4(7==  0  characterizes  the  Parabola  ;  \ 
£2  _  4^1  0<  o  characterizes  the  Ellipse  ;       I    (D). 
#2  —  4^4  0>  0  characterizes  the  Hyperbola.  ) 

Ex.  1.     Determine  the  species  of  the  locus  of  the  equation  2?/2  — 
5^2  —  2    —  12  =  0. 


SUG.—  As  the  equation  is  of  the  second  degree  the  locus  is  a  conic  section. 
Again,  in  this  case,  A  =  2,  B  ==  —  3,  and  6'=  5.  .  •.  £2  —  4AC  =  9  —  40  =  - 
31  <^  0  ;  and  the  locus  is  an  ellipse. 


Ex.  2.     Determine  the  species  of  the  locus  of  the  equation  y*  - 
5.ry  _  B.r2  +  2#  —  8  =  0.  The  locus  is  an  Hyperbola. 


OF  THE   CONIC   SECTIONS.  43 

Ex's  3  to  10.  Determine  the  species  of  the  loci  of  the  following 
equations  :  (1)  3y*  —  2a?«  —  4y  +  20  =  0.  (2)  4y*  —  2y  +  x  =  0. 

=  2(ar  +  3).     (6)  y  =  3(*  —  2).     (7)  y*  —  §a?  =  2(a?  —  2/)»  +  y, 

T/ie  l*tf  is  flw  Hyperbola ;  the  2nd,  a  Parabola ;  the  3rd,  an  Hyper 
bola;  the  4th,  an  Ellipse;  the  Bth,  a  Parabola;  the  Gth,  a  Parabola;  the 
1th,  an  Hyperbola. 

Suo's.-  Equations  like  the  4th  must  be  put  in  the  form  Ay"-  +  Bxy  -f-  Oc«  + 
j)y  +  Ex  +  F=Q,  before  applying  the  test.  Thus  equation  (4)  becomes  7/2  -f 
2ay -f- 5»*  —  2y  —  37x  +  100  =  0.  From  this  A  =  l,  5  =  2,  and  C  =  5.  .'. 
52  _  4.4(7=  —  1(3  <  0.  In  (3),  A  =  0,  B  =  4,  and  (7=  0. 

f>5.  COR.  I.— The  species  of  the  locus  depends  solely  upon  the  coeffi 
cients  A,  B,  and  C. 

04.  COR.  2.— The  form  Ay'  +  Cx*  +  Dy  +  Ex  +  F  =  =  0  embraces 
all  species  and  varieties  of  the  conic  section. 

DEM.  —Whatever  the  locus  may  be,  if  the  axis  of  abscissas  is  assumed  parallel 
to  the  axis  of  the  locus,  a  =  0.  Hence  sin  a  =  0  ;  and  B,  which  equals 
— 2e?  sin  a  cos  a,  is  0.  In  like  manner,  by  assuming  the  axis  of  abscissas  perpen 
dicular  to  the  axis  of  the  locus,  cos  a.  —  0  ;  and,  consequently,  B  =  0.  Q.  E.  D. 

65.     Prob.     To  determine  the  varieties  of  the  ellipse. 

SOLUTION.— As  the  equation  Atf*  -f  Ox*  -f  ^  +  &*  +  F  =  °  includes  all  spe 
cies  and  varieties  of  the  conic  sections  (04),  we  have  only  to  consider  what  loci  it 
represents  when  the  condition  B*  —  ±A  C  <  0  is  fulfilled.  This  condition  can  only 
be  fulfilled  when  A  and  C  have  like  signs  and  are  both  numerically  greater  than  0  ; 
for,  as  B  =  0,  if  A  and  C  have  different  signs,  B*  —  4, AC,  becomes  -f  4.4(7,  which 
is  greater  than  0.  If  A  or  C  =  0,  .B*  —  4  AC  =  0.  Now,  as  the  signs  of  A  and  C 
must  be  alike,  we  may  consider  them  as  always  -f  ;  because,  if  they  were  —  in  a 
given  case,  the  signs  of  all  the  terms  of  the  equation  could  be  changed.  Again, 
since  D  and  E  depend  upon  m  and  n  (D  =  2e?d  sin  a  —  2n,  and  E=  2e-d  cos  a  —  2m) , 
such  values  maybe  given  to  m  and  n,  i.  e.,  the  origin  may  be  so  located  with 
reference  to  the  focus,  that  D  and  E  shall  each  be  0  ;  nor  does  this  affect  the  spe 
cies  of  the  locus,  as  A  and  C  do  not  depend  upon  m  or  n.  Hence  we  learn  that  the 
form  Ay*  -f-  G»2  -f-  F  =  0,  embraces  all  the  varieties  of  the  ellipse. 

On  this  form  we  observe  that  if  .Fis  negative,  it  gives  by  transposition jty}  -f 

Cc*=F,  or  the  equation  of  the  ellipse  referred  to  its  axes,  in  which       I—    and 

I-   are  the  semi-axes  of  the  curve.     If  A  and   C  are  numerically  unequal  the 

axes  are  unequal,  and  we  have  the  Common  Ellipse.     If  A=  C,  the  axes  become 
equal  and  the  locus  is  a  Circle.      Again,   if    F=Q,  Ay*  -f  6^  =  0,  gives  y  =  d= 

—  -  a:*,  which  gives  no  real  values  except  x  =  0,  y  =  0  ;  and  hence  represents 
a  Point  (the  origin).     Finally,  if  Fis  -f,  in  the  form  Ay-  -f  6V-  -f-  F=0,  we  have 


44  THE   CARTESIAN   METHOD    OF   CO-OEDINATES. 


,  in  which  all  real  values  of  one  variable  give  imaginary  values 

to  the  other  ;  hence  the  equation  has  no  locus  in  the  plane  under  consideration. 

There  are,  therefore,  4  varieties  of  loci  embraced  in  the  equation  of  the  second 
degree  between  two  variables,  which  fulfill  the  condition  J32-—  4xlC<<0,  and  are 
hence  called  varieties  of  the  ellipse  ;  viz.,  the  Ellipse  proper,  the  Circle,  the  Point, 
and  the  Imaginary  locus. 


00.  I*rol>.   To  determine  the  varieties  of  the  hyperbola. 

SOLUTION. — Kesuming  the  equation  Ay*  -f-  Cx~  -}-  Dy  -\- Ex -{-  F=  0,  which  in 
cludes  all  species  and  varieties  of  conic  sections  (64),  we  observe  that  the 
characteristic  condition  of  the  hyperbola,  B-  —  4AC^>0,  can  only  be  fulfilled 
when  A  and  C  have  opposite  signs,  and  are  both  numerically  greater  than  0, 
since  B  =  0.  Also  that,  as  D  and  E  depend  upon  m  and  n,  and  A  and  C  do  not, 
such  values  may  be  given  to  m  and  n,  i.  e.,  the  origin  maybe  so  situated  with 
respect  to  the  focus,  that  D  and  E  shall  each  be  0.  Hence  Ay2  —  Cx~  -f-  F=  0, 
embraces  all  the  varieties  of  the  hyperbola. 

On  this  form  we  observe  that  if  Fis  positive,  it  gives  by  transposition  Ay2  —  Cx* 


=  —  F,  or  the  equation  of  the  hyperbola  referred  to  its  axes,  in  which       I -is  the 

J^_     Tfl 
is  the  semi-conjugate  axis.     If  A  and  G  are  numeri- 


I     ~ 
I  - 
\J       A 


cally  unequal  these  axes  are  unequal,  and  we  have  the  common  form  of  the  hyper 
bola.  If  A  =  C,  the  axes  become  equal  and  the  locus  is  an  Equilateral  Hyperbola. 
Again,  if  .Fis  negative  the  equation  becomes  Ay'2  —  Cx'2  =  F,  which  is  the  equation 
of  the  y  hyperbola,  since  the  real  axis  is  on  the  axis  of?/",  and  the  imaginary  one 


ou  the  axis  of  x.*     Finally,   if  F  =  0,  we  have  Ay*  —  Ox'2  =  0 ,  or  y  =  i      l-x, 
which  is  the  equation  of  two  straight  lines  passing  through  the  origin  and  making 

ni         i~c 

angles  with  the  axis  of  x,  whose  tangents  are  respectively      I  -r  and  —     I  —7. 

There  are,  therefore,  3  varieties  of  loci  embraced  in  the  equation  of  the  second 
degree  between  two  variables,  which  fulfill  the  condition  B-  —  4  A  C  ^>  0,  and  are 
hence  called  varieties  of  the  hyperbola  ;  viz.,  the  Hyperbola  with  unequal  axes, 
both  on  the  axis  of  x,  and  on  the  axis  of  y,  the  Equilateral  Hyperbola,  and  Two  Riyht 
Lines  intersecting  each  other. 


07 '•  JPTob,  To  determine  the  varieties  of  the  parabola. 

SOLUTION.  — As  the  equation  Ay-  -f-  C.v-  -f-  Dy  -\-Ex-\-  F=  0,  embraces  all  species 
and  varieties  of  the  conic  sections,  we  have  only  to  determine  what  loci  it  repre 
sents  when  B'2  —  4:AC=Q.  But  as  _Z>=0,  this  condition  can  only  be  fulfilled  by 


*  This  form  of  expression  is  frequently  used  instead  of  "  axis  of  ordinates,"  and  "  axis  of  abscis 
sas." 


OF   THE   CONIC   SECTIONS.  45 

A  =  0,  or  (7=  0.  Now  as  the  equation  is  symmetrical  with  respect  to  x  and  y,  it  will 
be  sufficient  to  examine  the  case  in  which  (7  =  0,  or  the  form  Ay2  -f-  Dy  -\-  Ex  -J- 
F=  0.  Remembering  that  a  has  been  made  0,  and  that  e  =  1,  we  find  D  (which 
equals  2e-d  sin  a  —  2n)  =  —  2n.  This  can  now  b«  made  0  by  taking  the  axis  of 
the  curve  for  the  axis  of  x,  and  the  equation  takes  the  form  Ay~  -j-  Ex  -f-  F  =.  0. 
But  in  this  case  we  cannot  make  E  =  (2e~d  cos  a  —  2m  =  2cZ  —  2m)  =  0,  since 
that  would  require  that  d  =  m,  which  is  absurd,  since  d  is  the  distance  from  the 
origin  to  the  directrix,  and  m  is  the  distance  from  the  origin  to  the  focus. 
(Numerically  d  may  equal  m  ;  but  E  =  0  requires  that  they  also  have  the  same 
sign). 

We  therefore  have  to  discuss  the  equation  Ay*  -f-  Ex  -f-  F==  0,  which  includes 
all  varieties  of  the  parabola.  As  .F  depends  upon  m-  -f-  n*  —  e~d-,  and  as  n  —  0, 
and  e  =  1,  ^Pniay  be  made  0,  by  putting  m  =  —  d,  which  only  requires  that  the 

origin  be  at  the  vertex.     The  equation  is  thus  reduced  to  y~  =  ±  —x,  the  d=  sign 

being  given  to  E,  as  no  restriction  has  been  imposed  upon  it.   This  is  the  common 

jfi 

equation  of  the  parabola,  in  which  —  =  2p.     The  -f-  sign  locates  the  curve  at  the 

A 

right  of  the  origin,  and  the  —  sign  at  the  left,  but  both  give  the  same  variety.  Again, 
if  in  Ay*  -\-  Bxy  -f-  GxP  -f  Dy  -f  Ex  -j-  F=  0,  we  make  A  =  0,  B  —  0,  and  C  =  0, 
the  condition  B-  —  4-4(7=  0  is  fulfilled,  and  the  locus  is  therefore  sometimes  called 
a  variety  of  the  parabola.  This  locus  is  evidently  a  right  line,  its  equation,  Dy  -j- 
E&  -f-  F=  0,  being  an  equation  of  the  first  degree  between  two  variables.  *  Finally, 
if  an  equation  of  the  second  degree  between  the  two  variables  can  be  reduced  to 
either  of  the  forms  y--}-  2yy-\-x-  ±  P(x  -f  y)  -f-  S  =  0,  y*  —  2xy  -f-  a2  db  P'  (x  —  y) 
-f  S'  =  0,  or  if-  ±  2x?/  +  x2  -f  8"  =  0,  the  condition  J3*  —  4.4(7=  0  is  still  fulfilled, 


although  the  equation  may  be  reduced  to  the  form  y  rb  x  =  m  iL  Vp  —  q,  which 
is  the  equation  of  two  real  or  imaginary  parallel  right  lines. 

We  have,  therefore,  4  varieties  of  the  Parabola  ;  viz.,  the  Common  Parabola,  the 
Bit/lit  Line,  Two  Parallel  Right  Lines,  and  Two  Parallel,  Imaginary  Right  Lines. 


OS.  COR.  1. — The  eccentricity  of  the  circle  is  0,  and  the  directrix  «« 
at  infinity. 

DEM. — In  obtaining  the  equation  of  the  circle  (65),  we  made  a  =  0,  and  A  =  C. 
Hence  1  =  1  —  e'2,  or  e  =  0.  Again,  when  the  ellipse  passes  into  the  circle,  the  foci 
unite  in  the  centre.  Now,  calling  the  distance  from  any  point  in  the  curve  to  the 

directrix  s,  and  the  radius  of  the  circle  R,  we  have  —  —  0  (the  distance  from  any 

s 

point  in  the  curve  to  the  focus   divided  by  its  distance  from  the   directrix  equals 
the  eccentricity)  ;  whence  s  =  cxO'  + 


*  In  reality  this  condition  is  not  compatible  with  our  fundamental  hypothesis,  which  requires 
the  equation  to  be  of  -the  second  degree.  Moreover,  the  conditions  A  =  0,  B  =  0,  and  (7=0,  are 
inconsistent  with  the  character  of  the  coefficients  A,  B,  and  C,  inasmuch  as  they  require  that 
1  —  e2sin2a=0,  1  —  e2cos'2o:  =  0,  and  2e2sin<*cos(*  =  0,  or  three  arbitrary  conditions  while  there  are 
but  two  arbitrary  constants,  e  and  (X. 

t  If  R  =  0,  the  condition  —  =  0,  is  satisfied  by  any  value  of  s.     In  this  case  the  locus  is  a 


46  THE   CARTESIAN   METHOD   OF   CO-ORDINATES. 

Ex.  1.  Determine  the  species  and  situation  of  the  locus  7/2  +  Gi/ 
—  12a?  +  33  =  0. 

SOLUTION.  —To  determine  the  species  of  the  locue,  observe  that  the  equation  is 
of  the  second  degree  between  two  variables,  and  fulfills  the  condition  B*—<iAC 
=  0.  Therefore,  the  locus  is  a  parabola. 

To  determine  the  situation  of  the  locus,  compare  its  equation  with  the  general 
equation  of  the  conic  section  :  viz., 

(1  —  e'-  sin2  «>'  —  2e*  sin  a  cos  a  xy  -f-  (1  —  en~  cos*  a)&  -f-  (2e*Z  sin  a  —  2n)y 
4-  (2e'-d  cos  a  —  2m).r  -f-  ?n*  -f-  na  —  e:d:  =  0. 

As  in  this  example  the  coefficient  of  ^  isl,  divide  the  general  equation  through 
by  1  -  e*  sin*  a  before  comparing  the  coefficients.     The  five  equations  from  which 
a,  e,  in,  n,  and  d  are  to  be  found  are  — 
m     -  2*  sin*  cos  « 


1  —  e«sins  a 

1-ei  cos^_  0     rl_,.^ns^  =  Q. 
v  '     1  —  e2  sin*  a 

^d^a-^n_  =      or  e,d  gin  a  _  n  =  3  _  3e5  Bin,  tt. 
1  —  e2  sin2  a 

2e*dcosa-2m  _    _12|  OT  e*d  <K«  a-m  =  -fr+6e*  sin'  a, 

1  —  e'2  sin'^  a 


1  —  e2  sin2  a: 

These  equations  are  now  to  be  solved  for  a,  e,  m,  n,  and  d.  But  as  the  locus  is 
a  parabola,  e  =  1.  Also  from  sin  a  cos  a  =  0,  a  must  be  either  0,  or  90°.  From 
(2)  cos  a  =  l.  .'.  a  =  0.  Making  these,  substitutions  in  (3),  (4),  and  (5),  they 
become  — 

(3,)  n  =  —  3  ;  (4,)  d  —  m  =  —  6;  (50  m2  -f  n2  —  d2  —  33.  These  equations  readily 
give  w  =  5,  and  d  =  —  1  . 

To  construct  the  locus,  let  XX  and  Y  Y  '  be  drawn  at  right  angles  to  each  other. 
Locate  the  focus,  F,  at  (5,  —3).  As  a  =  0,  draw  the  axis  of  the  locus  parallel  to 


point.    There  are  various  views  which  may  be  taken  of  the  eccentricity  of  a  right  line.    Thus,  consid 
ering  it  as  the  limit  of  the  hyperbola,  we  have  the  equation  y  = 

.  —  M 

^  I  -a;,    in   obtaining  which  we  put  F  —  m?  —  e^d-  =  0  (66). 

From  this  e  =  -.     But  m  =  Ae,  and  d  =  —     Hence  e  =  — -  =  e",        D 
d  e  A 


and  e=  1.      Therefore  y  =  \\-x  is  a  common  limit  of  the 

hyperbola  and  the  parabola.  Again,  if  we  may  be  allowed  to 
consider  Dy  +  Ex  +  F  =  0  (67)  the  equation  of  a  parabola,  wo 
reach  the  same  result  ;  viz.,  e  =  1.  If,  however,  we  consider 
C  B  tho,  directrix,  and  a  focus  removed  to  an  infinite  distance, 
and  MN  the  line,  PF,  the  distance  to  the  focus  is  GO,  and  FIG.  46. 

—  =  CO.    But  these  speculations  are  rather  curious  than  useful. 


F-f. 

/ 

G 


OF  THE   CONIC   SECTIONS. 


X'X,  and  through  F.  As  d  =  —  1,  take 
AG  =  —  1,  and  through  G,  drawing  C  B 
perpendicular  to  Z'Z,  it  is  the  direc 
trix.  Through  F  draw  QR  parallel  to 
CB  and  take  QF=  FR  =  FU.  The 
construction  can  now  be  completed  as  in 


FIG.  47. 


2.  Determine  the  species  and  situation  of  the  locus  y-  -f  %xy  -f 


SOLUTION.— Since  B*  —  4AC—  4  — 12  =  —  8  <  0,  the  locus  is  an  ellipse.     The 
five  equations  from  which  to  determine  a,  e,  d,  m,  and  n,  are — 


n        —  2e2  sin  a  cos  a 

~~ 


1  — 


2n 


=  '  or 


—  2m 


"We  observe  that  (1)  and  (2)  contain  only  the  unknown  quantities  e  and  a,  and 
hence  are  sufficient  in  themselves  to  determine  these  quantities.  From  (2)  we  get 
after  clearing  of  fractions,  substituting  for  cos2  a,  1  —  sin2  a,  and  reducing,  sin  a. 

The  -f-  sign  alone  is  given  to  the 

value  of  sin  a,  since,  by  reckoning  the  angle  from  0  to  180°,  we  get  all  possible  inch- 
nations  of  the  axis  of  the  locus  to  the  axis  of  reference.  Substituting  these  values  in 
(1)  and  reducing,  we  find  e  =  ,91-f-  ;  and  consequently,  sin  a  =  .9239,  and 
cos  a  =  rh  .3827.  .  •.  a  =  67°  30',  or  112°  30'.  To  determine  whether  cos  a  is 
+  or  —  ,  and  hence  whether  a.  is  67°  30',  or  112°  30',  we  consider  (1).  The 
denominator  of  the  first  member  being  necessarily  -{-,  since  1  >  e-sin2o:,  the 
numerator  must  be  rendered  -j-»  inasmuch  as  the  second  member  is  -}-.  But 
cos  a  is  the  only  factor  in  the  numerator  which  can  become  —  and  thus  render 
the  product  -f  .  .-.  cos  a  =  —  .3827;  and  a  =  112°  30'.  Substituting  the 
values  of  e,  sin  a  and  cos  a  in  (3),  (4),  and  (5),  they  become  (S^.n  =  .765d  ; 


(4t),  m  =  .586—  .317(Z;    and  (5,) 


n2  =  .82<Z*.     Substitutin     these  values 


of  m  and  n  in  (5,  ),  we  find  d  =  .72,  and  —  3.34. 
—  2.55.  '    m  =  .586  —  .317d  =  .36,  and  1.64. 


Finally  n  =  .765cZ  =  .55,  and 


THE  CARTESIAN  METHOD   OP   CO-ORDINATES. 


To  construct  the  figure,  draw 
G  G '  through  the  origin,  making  an 
angle  of  112°  30'  with  the  axis  of 
X.  Locate  the  foci,  F,  F',  at  (.36,  Q 
.55),  and  (1.G4,  —2.55).  Through 
one  focus,  as  F,  draw  a  line  ZZ' 
parallel  to  GG',  and  it  will  pass 
through  the  other  focus,  F',  if  the 
work  is  right.  Take  AG  =  .72 
and  AG'  =  —  3.34,  and  draw  CB 
and  C'B'  perpendicular  to  ZZ'; 
these  will  be  the  directrices.  Now 
the  ratio  e  =  .91,  the  focus  F,  and 
the  directrix  CB,  being  known, 
the  curve  can  be  constructed  (4:3}. 


FIG.  48. 

3.  Determine  the  character  and  situation  of  the  locus, 

=  0. 


SOLUTION.  — 7?2  —  ±AC=  4.  .  • .  The  locus  in  an  hyperbola.     The  five  equations 
from  which  e,  a,  m,  n,  and  d  are  determined  are — 

(1)  i^_^^  =0,  or  1  -e2  sin2  «  =  0  ; 

(2)  -ae'Bingcoaa  _  ^  sinacos  a  =TO. +„«-<*** ; 
W       m»_j_n*__c8(j* 

(3)  ~^~  —5^=0,  or  1  —  es  cos*  a.  =  0  ; 


(4)     ^^L=^J=0,  or<*Zsina-n  =  0; 

^cos^-Jm        _  d  CQS     _ 2     =     ^  _  wi,  _  ^ 

Vu/  ^.2     I     .,2          />2/7-2 


i 

The  general  equation  is  divided  through  by  m2  -f  n2  —  e2tZ^,  since  in  this  example 
the  absolute  term  is  1. 

From  (1)  we  have,  directly,  sin  a  =  -,  discarding  the  negative  root  for  the  rea- 

I         F  1    , -- 

son   given   in  the   preceding   solution.      .-.    cos  a  =±      |1 ^=   fc-ve8--!. 

Substituting  this  value  of  cos  a  in  (3),  we  have  e  =  \/%=  1.4142-f-.      Hence, 


or  135°.     Substituting  the  values  of  e  and  sin  a  in  (4),  we  have  n  =  \/td  ;  and 


OF  THE   CONIC   SECTIONS. 


N/     Y 


by  substitution  in  (2),  we  find 
m  —  zb  1.  In  this  case,  it  will 
be  seen  that  if  we  substitute  the 
-f-  value  of  cos  a  in  (2),  m  be 
comes  imaginary,  but  the  — 
value  gives  m  real.  Therefore, 

cos  a.  =  —  -  \/2,  and  a  =  135°. 
Finally,  substituting  in  (5),  we 

-1  O  

find  d  =  —  -r\/2,  and  j\/iJ,  or 

—  .35,    and   1.00  ;    and,    conse- 

1  3 

quently,  n  =  —  -,  and  -.      (The 

locus  is  situated  as  in  Fig.  49 ; 
but  the  construction  is  so  close 
ly  analogous  to  the  preceding, 
that  the  student  will  have  no 
difficulty  in  effecting  it.) 


FIG. 


4.  Determine  the  character  and  situation  of  the  locus  ?/2  —  2 xy 
+  2a?— 2a?=0, 

Results.     e=.924,   a  =  58°17',  d=  3.1  and  —.37,   m  =  1.78  and 
.21,  and  ?i  =  2.27  and  —.27. 

Sr-3. — In  problems  like  the  preceding  it  is  not  admissible  to  divide  the  general 
equation  through  by  a  coefficient  corresponding  to  one  which  is  0  in  the  partic 
ular  case,  inasmuch  as  this  process  would  reduce  each  coefficient  to  infinity  or  inde- 
termination. 

Thus  for  Ex.  3,  should  we  put  the  general  equation  in  the  form 
2e?  sin  a  cos  a       .    1  —  e2  cos*  a          2e2<Z  sin  a  —  2n 

y* '  -  -ir=-^^rxy + \=#ti&ax +-i=*^ry  4 

each  of  the  coefficients,  when  the  application  was  made  to  the  equation  Ixy  —  x 
-j-  1  =  0,  would  be  infinite  or  indeterminate,  since  in  this  example  1  —  e2  sin2  a. 
=  0. 


EXERCISES. 

[NOTE.— This  list  of  exercises  is  designed  to  give  the  student  an  opportunity  for  making  an  effort 
to  produce  the  equations  himself.  Nothing  new  is  developed  in  them,  and  the  student  need  not 
necessarily  tarry  till  he  has  mastered  them  all,  though  by  doing  so  clearness  and  breadth  of  view 
will  be  promoted.  Let  every  one  understand,  however,  that  ability  to  investigate— to  reason  for 
himself— is  the  proper  object  toward  the  attainment  of  which  he  should  strive.] 

1.  To  produce  the  equation  of  the  ellipse  referred  to  its  own  axes 
and  in  terms  of  its  semi-axes,  directly  from  the  definition,  with 
out  first  obtaining  the  general  equation  of  the  conic  section. 


60  THE   CARTESIAN  METHOD   OF  CO-ORDINATES. 

SUG'S.     AD=o-,    PD=y,    AF  =  Ae,  IV 

PF  c 

and  — —  =  e.     P  F2  =  ea  X  P  E2.     P  F2  = 

/A          \2         E 

2/2  -j-  (Ae  +  xY,    and    PE2  =  (^  -f  x)  . 

.-.  y2  •+•  (1  —  e«)ar*  =  ^12(1  —  e2).     For  1  —  e2 

B'- 
substituting    — ,  we    have   A-y*  -j-  JB2a;2  = 

B 

FIG.  50. 

2.  In  like  manner  produce  the  equation  of  the  ellipse  referred  to 
its  transverse  axis  and  a  tangent  at  its  left  hand  vertex,  i.  e.,  y2  = 


3.  Show  that  the  equation  of  an  ellipse  referred  to  its  conjugate 


A2 


axis  and  a  tangent  at  the  upper  vertex  thereof  is  a;2  =  --  (2By  +  y2). 

* 


•  PF 
SUG'S.    A  D  =SB,  P  D  =  —y,  —  =  e,  and      C 

PF2  =  e2  X  "PE2.     Also  "PF*  =  (a; -f  Ae}*   XjT 
•  y)2,  and  PE2  =  (  - — t- 


4.  Produce  the  common  equation 
of  the  hyperbola,  A-ij9  —  jB2ar2  =  — 

AZB2,  directly  as  above. 

FIG.  51. 

5.  In  like  manner  as  above  produce  the  common  equation  of  the 
parabola,  y-  = 


6.  Show  that  the  equation  of  an  ellipse  iaA*(y  —  y^"  +  R*(x  —  x^* 
=  A*B\  when  a-v  and  yl  are  the  co-ordinates  of  the  centre,  and  the 
axes  of  reference  are  parallel  to  the  axes  of  the  curve. 


SUG'S.     AL=Bi,   OL  ==t/i,  AD  =x, 
PD  =  y,   and  "PF2  =  e*  X  PEX      Also  H 
"PF2=PR2-f  FR2>  PR=?/  —  y},  FR 

=  x  —  O5j  +  Ae,  and  PE  =  PH  —  HE 


=  0;  —  Xj   -j  --  .     Siibstituting,    (y  — 


-f-  x,2  — 


-f 


-j-  c2.^2  — 


Transposing  and  collecting  terms, 


OF   THE   CONIC   SECTIONS. 


51 


p~ 

=  ^T2(l  —  e-).     Putting  —  for  1  —  e2,  we  have 


(y  _  yi  )2  -f-  (1  —  e2)a2  —  (1  — 
(y  __  yi  )2  -f  (1  —  ea)(x  —  .^O2 

^3<y  -  2/1  )2  +  ^2(*  -  »i)2 

7.  Deduce  from  the  general  equation  of  the  conic  section  (SI)  the 
equation  of  a  parabola  whose  parameter  is  2p,  referred  to  rectangular 
axes,  the  axis  of  the  curve  falling  on  the  axis  of  abscissas,  and  the 
vertex  of  the  curve  being  at  ^  to  the  right  of  the  origin.  Also  the 
equation  when  the  vertex  is  at  ^  to  the  left  of  the  origin.  Also  the 
equation  when  the  vertex  is  at  (x^  y{). 

The  equations  are  y*  =  2p(x  —  xj,  y*  =  2p(x  +  #1),  and  (y  —  y,)*  = 


69.  GENERAL  SCHOLIUM.— It  will  appear 
hereafter  that  the  conic  sections  are  formed  by 
the  mutual  intersection  of  a  plane  and  a  right 
cone  with  a  circular  base.  It  is  from  this  fact 
that  the  name  of  these  curves  is  derived  ;  and 
from  this  as  a  definition  they  were  formerly 
studied.  The  different  species  arise  from  dif 
ferent  positions  of  the  cutting  plane.  The 
plane  which  gives  the  parabola  lies  parallel  to 
one  of  the  elements  of  the  cone,  as  ON  P,  and 
hence  cuts  but  one  nappe,  and  gives  but  one 
branch.  To  produce  the  ellipse  the  cutting 
plane  lies  between  this  position  and  perpendic 
ular  to  the  axis,  as  RTmn.  To  produce  the 
hyperbola  it  lies  between  parallel  to  an  element 

and  parallel  to  the  axis,  as    HIK    and  EGF,  and  hence  cuts  both  nappes, 
giving  two  branches. 

Several  of  the  varieties  of  the  conic  sections  as  heretofore  considered,  may 
bo  illustrated  by  means  of  this  geometrical  conception,  and  their  mutual 
relations  more  clearly  seen.  Thus  as  the  plane  of  the  ellipse  approaches 
perpendicularity  to  the  axis,  the  ellipse  approaches  the  form  of  a  circle  into 
which  it  passes  when  the  plane  becomes  perpendicular  to  the  axis.  The 
circle  is  therefore  a  variety  (or  more  properly  a  limit)  of  the  ellipse.  So 
also  as  the  plane  approaches  the  vertex,  the  ellipse  diminishes,  passing  into 
its  limit— a  point— when  the  plane  passes  through  the  vertex.  The  hyper 
bola  becomes  two  intersecting  straight  lines  when  the  cutting  plane  passes 
through  the  vertex  and  is  not  parallel  to  an  element.  When  it  becomes 
parallel  to  an  element  and  also  passes  through  the  vertex,  it  gives  the  limit 
both  of  the  parabola  and  the  hyperbola,  which  common  limit  is  a  right 
line,  When  the  cone  passes  into  a  cylinder  the  parabola  becomes  tAvo  par 
allel  right  lines.,  as  also  the  hyperbola  may,  if  it  is  conceived  as  produced  ^by 
a  cutting  plane  perpendicular  to  the  base.  If  the  cutting  plane  producing 


52  THE   CARTESIAN  METHOD   OF   CO-ORDINATES. 

the  hyperbola  is  conceived  as  oblique  to  the  axis,  the  hyperbola  passes  into 
an  ellipse  when  the  cone  passes  into  a  cylinder. 

[NOTE.— Of  course  the  above  views  are  not  give-i  as  in  any  sense  needed  to  confirm  the  conclu 
sions  of  the  preceding  discussions,  but  simply  to  give  the  student  a  little  further  insight  into  the 
wonderful  harmony  which  exists  between  algebraic  formula  and  geometrical  loci.] 


Through  Jive  points  in  a  plane  one  conic  section  may 
always  be  made  to  pass,  and  but  one. 

DEM.— Dividing  the  general  equation  Ay"--\-  Bxy-\-Cx2  -\-Dy-\-Ex-\-  F=  0  through 
by  F,  and  distinguishing  the  new  coefficients  by  accents,  we  have  Ay-  +  B'xy 
+  C"^  +  #'2/  +  A"x-+l=0.  Now  let  (xlt  ?/,),  (;c2,  y2),  (a3,  y.A\  (x,,y,),  and 
(x5,  7/3)  bo  the  live  given  points.  Substituting,  successively,  in  the  last  equation, 
the  co-ordinates  of  these  five  points,  for  .the  general  co-ordinates  x  and  y,  there 
result  the  five  equations 

A'y^  +  E'x^  +Vxl*  +  D'yi  +  Ex,  +1=0; 

A'y**  +  B'x,y.z  +  CxS  +  D'y.2  +  Ext  +1=0; 

A'yj  +  B-x*y*  +  ffx^  +  D'y,  +  E'x,  +  1  =  0  ; 

A'yj  +  E'Wi  +  C'xj  +  D'y,  +  E'x.  +  1  =  0; 

A' y6*  +  B'x6y6  +  C'x^  +  D'y^  +  Exs  +  1  ==  0  ;  or  five  arbitrary  condi 
tions.  This  number  of  conditions  is  possible,  since  there  are  five  arbitrary  constants 
involved  ;  viz.,  A,  B ',  C',  D',  and  E'.  From  these  equations,  asic,,  7/l5  jc2,  t/2, 
.T3,  2/:J,  a;4,  ?/4,  a;.-,,  and  ys,  are  known  quantities,  the  values  of  A',  B ,  C',  D',  and  E" 
can  be  determined.  Having  found  the  values  of  these  coefficients,  by  substituting 
their  values  in  the  general  equation  A'y*  +  B'xy  +  (7z2  +  D'y  -\-Ex  +  1  =  0, 
there  results  an  equation  of  the  second  degree  between  two  variables,  or  an  equa 
tion  of  a  conic  section.  As  this  equation  is  satisfied  by  the  co-ordinates  of  each  of 
the  five  given  points,  the  locus  represented  by  them  passes  through  these  points. 
Finally,  as  the  five  equations  are  all  of  the  first  degree  with  respect  to  A',  B',  C', 
D',  and  E',  but  one  set  of  values  can  be  determined  for  these  coefficients.  There 
fore,  but  one  conic  section  can  be  made  to  pass  through  the  five  given  points.  Q.  E.  D. 

SCH.  1. — In  this  proposition  the  term  Conic  Section  must  be  taken  in  its 
broadest  sense,  i.  e. ,  as  embracing  all  varieties  of  these  loci,  except  the  so- 
called  imaginary  loci. 

SCH.  2.  — If  the  five  points  are  so  situated  that  the  equation  of  the  locus 
passing  through  them  lacks  some  of  the  terms  of  a  complete  equation,  it  will 
not  do  to  divide  the  general  equation  by  the  coefficient  of  such  a  term.  If 
such  an  error  has  been  made  in  the  hypothesis  in  any  solution,  it  will  soon 
appear  as  the  solution  proceeds.  This  case  is  analogous  to  the  one  noticed 
in  the  suggestion  under  Ex.  4,  page  49. 

Ex.  1.  Produce  the  equation  of  a  conic  section  passing  through  the 
five  points  (2,  3),  (0,  4),  (—1,  5),  (  —2,  —1),  and  (1,  —2),  and  deter 
mine  its  species. 

SOLUTION. — The  five  equations  which  determine  the  coefficients  A',  B',  C',  D't 
and  E',  are 


OF   THE    CONIC    SECTIONS.  53 

(1)  94'  -f  61?'  +  46"  -f  3D'  +2#-fl=0; 

(2)  164'  +  4i>'  +1  =0; 

(3)  '25 A'  —  55'  +    6"  -f  5Z/  —    ^  +  1  =  0  ; 

(4)  A'  4-25'  4-  4(7  —   I/  —  2tf  4-1  =0  ; 

(5)  44'  —25'  +    6''  —  25'  4-    #4-1=0. 

169  220  89  445 

Solving  these  equations,  we  find  A'=    —  — ^  JJ   =    —&£'  C    ^  92T'        =  924~' 

113 

JET  =  rT-^-.     Substituting  these  values  in  the  general  equation  Ay*  +  B'xy  -}-  <7x2 

4-5'7/-'4- JS"x  4-1=0,  clearing  of  fractions  and  changing  signs,  we  have  169?/- 
-f  220xr/  —  89x-  —  445?/  +  113x  —  924  =0,  which  is  the  equation  of  a  conic  section 
passing  through  the  five  given  points.  This  locus  is  an  hyperbola,  since  52  —  44(7 
>0. 

Ex.  2.  Produce  the  equation  of  a  conic  section  passing  through 
the  live  points  (1,  3),  (4,  —6),  (0,  0),  (9,  —9),  and  (10, 12),  and  find 
its  species. 

SUGGESTIONS.— As  one  of  the  given  points  is  (0,  0),  the  locus  passes  through  the 
origin  ;  and  hence  F=0.  The  form  of  the  general  equation  used  would,  there 
fore,  be  Ay*  -f  Bxy  -f-  Cx-  4-  Dy  -\-  Re  =  0,  which  divided  through  by  one  of  the 
coefficients,  as  A,  gives  the  form  y-  -\-£'xy -}-C'x--+-D'y  +  #x=0.  This  equa 
tion  satisfied  for  the  four  points  (1,  3),  (4,  —6),  (9,  —9),  and  (1G,  12),  in  succession, 
gives  rise  to  four  equations  from  which  the  coefficients  can  be  determined. 

The  locus  is  a  parabola  ivhose  equation  is  y2  =  9x. 

Ex.  3.  Produce  the  equation  of  a  conic  section  passing  through 
(_4,  _2),  (2,  1),  (—6,  3),  (0,  0),  and  (2,  —1),  and  determine  its 
species.  The  equation  is  y  =  qp  Ja?. 

Ex.  4.  Produce  the  equation  of  a  conic  section  passing  through 
(3j  v/5)}  (_2,  Q),  (—4,  —  v/12),  (3,  —  \/5),  and  (2,  0),  and  deter 
mine  its  species.  The  locus  is  an  equilateral  hyperbola. 

Ex.  5.  Produce  the  equation  of  a  conic  section  passing  through 
(—&  —  i)>  (2,  1),  (f>  2),  (— f,  —3),  and  (f,  —  $)  and  determine 
its  species. 

T/ie  /ocws  is  an  ellipse  whose  equation  is  y~  —  %xy  -\-  ox-  4-  2y  —  4# 
-3  =  0. 

Ex.  6.  What  is  the  equation  of  a  circle  whose  radius  is  5,  referred 
to  rectangular  axes,  and  the  origin  at  the  centre  ?  When  the  origin 
is  on  the  circumference  and  the  axis  of  abscissas  is  a  diameter? 
When  the  axes  are  tangent  to  the  circumference  ? 

Equations,  y*  +  x*  =  25,  y2  =  ±  IQx  —  x2,  and  y*-  +  xz  —  10y  —  lOx 
-f  25  ==  0. 

Ex.  7.  What  is  the  equation  of  an  ellipse  whose  axes  are  16  and 
10,  when  referred  to  its  own  axes  ?  When  referred  to  its  transverse 


54 


THE  CARTESIAN   METHOD  OF  CO-ORDINATES. 


axis  and  a  tangent  at  the  left  hand  vertex  ? 
lems  in  the  case  of  the  hyperbola. 

Equations,  64y2  +  25x*  =  1600. 

=  _  1600.     6 


The  corresponding  prob 


—  400^  -f 
400^  — 


=  0. 
=  0. 


SUGS.  —  The  results  of  the  two  preceding  examples  are  readily  written  from  the 
equations  of  the  respective  loci  as  given  in  (53  —  57)>  and  are  designed  to  fami 
liarize  those  most  important  forms. 

Ex.  8.  Produce  the  equation  of  a  parabola  referred  to  rectangular 
axes,  the  vertex  of  the  parabola  being  at  (  —  3,  —  2),  the  parameter,  6, 
and  the  axis  of  abscissas  parallel  to  the  axis  of  the  curve. 

Equation,  y*  +  4?/  —  Gx  —  14  =  0. 

Ex.  9.  Produce  the  equation  of  an  ellipse  whose  eccentricity  is  §, 
its  major  axis  18,  the  centre  being  at  (  —  2,  3),  and  the  axes  of  refer 
ence  being  rectangular  and  parallel  to  the  axes  of  the  curve. 

Equation,  9y2  +  5#2  —  54?y  -f  20#  —  304  =  0. 

Ex.  10.  What   are   the  following  loci,  and  what  their  axes  :  viz., 

36?     7^  +  I!*/2  =  15?     !(%'  —  25#«=  —  2,500? 
=  —  116? 


EXERCISES    IN    PRODUCING    THE   EQUATIONS   OF  THE   CONIC   SECTIONS 
FROM   OTHER   DEFINITIONS. 

[NOTE.  —  These  exercises  may  be  omitted  without  destroying  the  integrity  of  the  course.  They 
are  designed  simply  to  lead  the  student  to  a  more  full  comprehension  of  the  process  of  producing 
an  equation  of  a  locus  from  its  definition,  a  subject  of  vital  importance  if  one  proposes  to  so  master 
this  method  of  geometrical  investigation  as  to  be  independent  in  the  use  of  it.] 

1.  To  produce  the  common  equation  of  the  ellipse  from  the  defini 
tion  :  —  The  ellipse  is  a  curve  such  that  the  sum  of  the  distances  from  any 
point  in  the  curve  to  two  fixed  points  called  the  foci,  is  constant  and  equal 
to  the  major  diameter. 

SUGS.       AD=.r,    PD=?/,    A  B  =  A, 
A  E  =  B,  A  F  =  A  F'  =  c.    Then  from  the 


definition  Vy"~  +  (c  +  x)*  +  \/V  +  i  c  —  x)2 
=  2 A.  Whence  A*y*  +  (A*—  c2)x2  =  A* 
(A*  —  c2).  But  by  definition,  E  F  =  A  , 
whence  A'2  —  c2  =  B2  ;  and  we  have  A-y'2 


2.  In  a  manner  similar  to  the  above 
produce   the   common    equation    of  FIG.  54. 

the  hyperbola,  from  the  definition,—  The  hyperbola  is  a  curve  such 
that  the  difference  of  the  distances  from  any  point  in  the  curve  to  two  fixed 
points  is  constant  and  equal  to  the  transverse  axis. 


OF   THE    CONIC    SECTIONS. 


55 


gUG's. ln  this  case  it  must  be  borne  in  mind  that  Az-\-  B'2=c^  (47),  and  hence 

that  A2 c'2  =  —B~.     The  equation  is  Azy*  —  _B2x2  =  — A*B2,  as  before  produced. 


3.  To  produce  the  equation  of  the  locus  of  a  point  moving  so  that 
the  square  of  its  distance  from  a  fixed  point  is  in  a  constant  ratio  to 
its  distance  from  a  fixed  line. 


SUG'S. — Let  the  fixed  line  be  taken  as  the 
axis  of  abscissas,  and  let  a  perpendicular  to 
it  through  the  fixed  point,  F,  be  taken  as  the 
axis  of  ordinates.  Let  P  be  any  point  in  the 

locus.      Then   AD=.r,    and   PD=y.      As      

A  F  is   constant,   cull  it  a,   and   let   w   repre-      X 

sent  the  ratio  referred  to  in  the  definition  in  the 

example.     The  equation  sought  is  (y  —  a)'2  -f-  x'2  Y' 

=my,  or  y2  -f-  ar*  —  (2«  -|-  m)y  -f-  a2  =  0.     This  FIG.  55. 

being  an  equation  of  the  second  degree,  the  locus  is  a  conic  section.     Again,  as 

jj-2  —  4  j[  C  <^Q,  it  is  an  ellipse.     Finally,  as  the  coefficients  of  y2  and  x-  are  equal, 

it  is  an  ellipse  with  equal  axes,  or  a  circle. 

To  determine  more   fully  the   situation  of  this    circle,  notice   that   for  y  =  0, 
x  =  ^  \/ a2,  whence  we  see  that,  in  general,  the  circle  does  not  cut  the  axis  of  x. 


Making  x  =  0,  y  = 


2*7 -fm 


4mxi  +  in'2       .._ 

.     ISow,   as   every  value  of  ?/  in   the 


~\J 

equation  of  this  locus  gives   two  values  of  x,  numerically  equal  but   with  oppo 
site  signs,  we  see  that  the  locus  is  symmetrical  with  the  axis  of  y,  and  that  the  cen- 

2  a  4-  m 
tre  of  the  circle   lies   in   this   axis.     But   the   circle   cuts   this   axis   at 


•f 


-f-  ?»ia 


L. — ".     Whence   the    diameter   is  the 


difference    between    these    values  ;     and    letting    r    be    the     radius,     we     have 

r  =  r\/±ay/i  -j-  m-,  L  e. ,  the  radical  part  of  the  root. 

2 

In  the  particular  case  in  which  a  =  0  ;  L  e.,  when  F  is  at  A,  the  equation  be 
comes  y-  4-  x1  =  my,  which  is  the  equation  of  a  circle  passing  through  the  origin, 
and  having  its  centre  on  the  axis  of  y. 


4.  In  the  given  right  lines  A  P,  A  Q,  intersecting  at  right  angles, 
are  taken  variable  points  p,  q,  such  that  Ap  I  p  P  :  :  Q/?  *  #A  ; 
prove  that  the  locus  of  the  intersection  of  Pq,  Qp,  is  an  ellipse  which 
touches  the  right  lines  in  P  and  Q. 


56 


THE   CARTESIAN   METHOD    OF   CO-OKDINATES. 


D    P 


FIG.  56. 


SUG'S.—  Let  AP  and  AQ  be  taken  as  the  axes 
of  reference.  Call  AP  =  a,  and  AQ  —  6.  Then 
A  D  =  x,  and  R  D  =  y.  From  the  similar  trian 
gles  RPD,  q PA,  and  RpD,  QpA  obtain  the 
relation  between  x,  y,  a,  and  b.  This  will  be  the 
equation  sought.  It  is,  when  reduced,  a~(y — 6)2 
-j-  b'-(x  —  a)'2  =  a'262  —  abxy. 

SCH.  — As  this  equation  is  of  the 
second  degree,  and  B*  —  4^4(7  = 
—  3a262,  the  locus  is  an  ellipse. 
As  there  is  a  term  in  scy,  the  axis 
of  the  curve  is  inclined  to  the 
axis  of  abscissas.  For  x  =  a, 
y  =  b  and  0  ;  hence  the  locus 
passes  through  (a,  b),  and  P. 
For  x  —  0,  y  =  b  ;  therefore  the 
locus  passes  through  Q. 

To  effect  the  construction  mechanically,  take  Aj9  :pP  ::  Qq  :  ^A,  by 
composition  and  alternation,  giving  AP  :  AQ  :  :  Pp  :  A?.  Now  assuming 
p  at  any  point  in  ax,  we  can  find  the  corresponding  value  of  A#.  After 
p  passes  P,  A?  becomes  — ,  and  is  laid  off  below  A.  So  when  Qpf  passes 
parallelism  with  AX,  kp  becomes  negative. 

5.  Required  the  locus  of  the  middle  point  of  a  line  moving  with  its 
extremities  in  two  fixed  lines  at  right  angles  with  each  other,  while 
it  passes  through  a  fixed  point. 

SUG'S.  — Take  the  fixed  lines  as  axes  of  refer 
ence.  Let  O  be  the  fixed  point,  and  C  B  the 
line,  the  locus  of  whose  centre,  P,  is  to  be  de 
termined.  Calling  AD  a,  and  OD  b,  the 
equation  of  the  locus  is  2a-y  —  ay  —  bx  =  Q, 
which  is  the  equation  of  an  hyperbola,  passing 
through  the  origin,  since  for  x  =  Q,  y  =  0;  and 
also  passing  through  O,  since  x—a,  gives  y 
=  6.  Let  the  pupil  trace  the  curve. 


FIG.  57. 


6.  Required  the  locus  of  the  point  P, 
moving  so  that  P  D  ,  bears  a  constant 
ratio  to  A  D  X  D  B  ;  A  and  B  being 
fixed  points.  What  is  the  locus  when 
this  ratio  is  1  ? 


x 


The  distance    A  B  being  catted  2a,  and  FIG.  58. 

the  ratio  m,  the  equation  is  y2  =  mx(2a  — x),  whence  the  locus  is  seen 
to  be  an  ellipse.     Jfm=  1,  it  is  a  circle. 


EQUATIONS  OF  HIGHER  PLANE  CURVES.  57 

7.  If  P- moves  in  Fig.  58,  so  that   PD*  bears  a  constant  ratio  to 
A  D,  what  is  the  locus? 


SECTION  VII. 
Equations  of  Higher  Plane  Curves, 

71.  One  variable  is  called  a  Function  of  another  variable  when 
it  depends  upon  that  other  variable  for  its  value.     Thus  the  ordinate 
of  a  curve  is  a  function  of  the  abscissa. 

72.  Functions  are  classified  as  Algebraic  and  Transcen 
dental;   and  the  latter   are  subdivided  into  Trigonometric, 
and  Circular,  Logarithmic,  and  Exponential. 

73.  An  Algebraic  Function  is  one  which  involves  only  the 
elementary  methods  of  combination,  viz.,  addition,  subtraction,  mul 
tiplication,  division,  involution  and  evolution.    Thus  in  y  =  ax*  • —  3#3, 
and  in  all  the  equations  hitherto  discussed  in  this  chapter,  y  is  an 
algebraic  function  of  xt  except  24-33,  Sec.  II. 

74.  A  Trigonometrical  Function  is  one  which  involves 
sines,  cosines,  tangents,  cotangents,  etc.,  as  y  =  sin  x,  y  —  sin  x  tan 
x,  etc. 

7 «£.  A  Circular  Function  is  one  in  which  the  concept  is  an 
arc  (in  the  trigonometrical  the  concept  is  a  right  line).  These  are 
written  thus  :  y  =  sin~lx,  read  "y  equals  the  arc  whose  sine  is  #"; 
2/=tan~X  read  "y  equals  the  arc  whose  tangent  is  x." 

Notice  that  in  the  expression  y  =  tan~'#,  it  is  the  arc  which  we  are 
to  think  of,  while  in  the  expression  x=t&n.y  it  is  the  tangent,  which 
is  a  right  line.  Trigonometrical  functions  are  right  lines  ;  circular 
functions  are  arcs.  These  functions  are  mutually  convertible  into 
each  other ;  thus,  y  =  sin~lx,  is  equivalent  to  x  =  sin  y,  the  only 
difference  being  that  in  the  former  we  think  of  the  arc,  the  sine  being 
given  to  tell  what  arc,  and  in  the  latter,  we  think  of  its  sine,  the  arc 
being  given  to  tell  what  sine. 

The  circular  functions  y  =  siii"1^,  y  =  cos"1^,  y  =  sec~X  etc.,  are 
often  called  the  Inverse  Trigonometrical  Functions. 

70.  A  Logarithmic  Function  is  one  which  involves  loga 
rithms  ;  as  y  =  log  xt  log2  y  =  3  log  ax,  etc. 

77.  An  Exponential  Function  is  one  in  which  the  varia- 
able  occurs  as  an  exponent ;  as  y  —  ax,  z  =  xv,  etc. 


58 


THE   CARTESIAN   METHOD    OF   CO-OKDINATES. 


78.  Hiylier  Plane  Curves  are  loci  whose   equations  are 
above  the  second  degree,  or  which  involve  transcendental  functions. 
As  it  has  already  been  shown  that  loci  of  the  equations  of  the  1st 
degree  are  right  lines,  and  that  loci  of  the  2nd  degree  are  conic  sec 
tions,  it  follows  that  all  other  plane   loci  are  higher  plane  curves. 
The  former  are  called  Lower  Plane  Loci. 

Of  course  the  variety  of  higher  plane  loci  is  infinite.     We  can  con 
sider  but  a  few,  and  these  simply  as  specimens. 

79.  An  Algebraic  Curve  is  one  whose  equation  contains  only  alge 
braic  functions.     A  Transcendental  Curve  is  one  whose  equation  con 
tains  transcendental  functions  ;  when  converted  into  algebraic  forms 
their  degree  is  infinite. 


THE   CISSOII)   OF   D10CLES. 

80.  DEF. — If  pairs  of  equal  ordinates  be  drawn  to  the  diameter 
of  a  circle,  and  through  one  extremity  of  this  diameter  and  the  point 
in  the  circumference  through  which  one  of  the  ordinates  is  let  fall,  a 
line  be  drawn,  the  locus  of  the  intersection  of  this  line  and  the  equal 
ordinate,  or  that  ordinate  produced,  is  the  Cissoid  of  Diodes'. 

SI.  I*rob.     To  construct  the  Cissoid. 

SOLUTION.— Let  AB  be  the  diameter  of  a  circle  ;  and  ED,  and 
E'D'  be  equal  ordinates.     Through  A  and  E'  draw  AE'  inter 
secting  ED  in  P.     Then  is  P  a  point  in  the  locus.     In  like 
manner  draw  AE  and  produce  it  till  it  meets  E'D' 
produced  in  |.     Then  is  |  a  point  in  the  locus.     In 
the  sumeVay  other  points  are  found  both  above  and 
below  AB.     There  are,  therefore,  two  branches  of  the 
locus  ACM  and  AC'M',  symmetrical  with  respect 
to    the    diameter    AB-      These    branches    evidently 
meet  at   A,  pass   through  the  extremities  of  the  di 
ameter  CC',   and  have   GG'  as  a  common  asymp 
tote. 

Sen.  1. — The  name  Cissoid  is  from  the  Greek 
and    signifies  ivy-form.     It  was    applied   to    the 
curve,    probably,    from    its   resemblance    to    the 
graceful   outline    presented  by  &  growth  of   ivy 
upon  a  wall.     The  locus  was  invented  by  the  Greek 
geometer    whose    name  it  bears,   while    he  was  seeking 
the  solution  of  the  celebrated  problem  of  the  Duplication 
of  the  Cube. 

Sen.  2. — Sir  Isaac  Newton  gave  the  following  mechan 
ical  method  of   describing  this    locus  :   Let   AB   be  the 


EQUATIONS  OF  HIGHER  PLANE  CURVES. 


59 


diameter  of  the  circle  from  which  the  curve  would  be 
described  by  the  definition  ;   at  the  centre  O  erect  the 
perpendicular  OL»  and  take  AD  =  AO  =  OB-     Now 
take  a  rectangular  ruler  FEC,  whose  leg  CE  =  AB,  and 
while  the  extremity  C  moves  in  the  line  OL,  let  the  leg 
FE  slide  through  the  fixed  point  D,  then  will  the  middle 
point  of  CE,   P,   describe  the  cissoid.      [The 
demonstration  will  afford  a  good  exercise  for  the 
student.  ] 


Ll     M,    G 


SCH.  3. — This  curve  is  also  the  locus  of  the 
vertex  of  a  common  parabola  rolling  upon  an 
equal  parabola. 


FIG. 


82.  I*rob* — To  produce  the  equation  of  the  Cissoid  of  Diodes. 

SOLUTION.— In  Fig.  59  let  AX  and  AY  be  the  axes  of  reference,  AB  =  2a, 
the  diameter  of  the  circle  referred  to  in  the  definition,  and  P  any  point  in  the 
curve.  Then  AD  =  x,  and  PD  =y.  Draw  through  P  the  ordinate  ED,  and 
also  draw  the  equal  ordinate  E  D'.  APE'  is  a  straight  line  by  definition.  We 
now  have  AD  :PD  ::AD'  :  E'  D',  or  x  :  T/  : :  2a  —  x  :  v/(2a  —  x)x.  Squar- 

x* 
ing  and  reducing,  xz  :  y2  : :  2a  —  x  :  x.     .  • .  y*  =  — '- is  the  equation  sought. 


SCH.  1.—  Since      =  ± 


-,  every  real  -f  value  of  x  <  2a  gives  two  real 


and  numerically  equal  values  of  y,  with  contrary  signs.  Hence  the  locus  is 
symmetrical  with  respect  to  the  axis  of  x.  For  x  =  2a,  y  =  ±  GO  ,  whence  the 
branches  are  infinite,  and  GG'  is  an  asymptote  to  both  branches.  For  all 
values  o/"x>>2a,  and for  x  negative,  y  is  imaginary.  TJierefore  the  locus  is 
comprised  between  the  limits  x  =  0,  x  =  2a. 


To  effect  this 

L 


SCH.  2.  —  By  the  Duplication  of  the  Cube  is  meant  finding  the  edge  of  a  cube 
which  shall  hqve  twice  the  volume  of  a  cube  whose  edge  is  given. 
by  means  of  this  curve,  let  A  M   be  any  cissoid,    A  B  the 
diameter  of  the  circle  which  pertains  to  it,  and  O  the  centre 
of   that  circle.     Take  CO  =2  OB,   and  draw  CB.      Let 
fall  from  the  point  P,  where  C  B  cuts  the  curve,  the  per 
pendicular    PK.     Then    PK  =  2BK.      Now  a  cube   des 
cribed  on  P  K  is  twice  one  described  on  A  K  ;  for 
since  P  K  =  y,  A  K  =  x,  and  K  B  =  2a  —  a;,  we  have 


=  2AK'.  Finally,  let  a  be  the  edge  of  any  given 
cube  ;  find  «,  so  that  a:cti  ::  AK  :  PK,  whence  a3  : 
P~Ka  =  2AK8.  .'.  a,3  =  2a3. 


:  A  Ks :  PK3.      But 


60 


THE   CARTESIAN   METHOD    OF   CO-ORDINATES. 


By  taking  CO  —  SOB  and  proceeding  in  a  similar  manner,  we  can  tri 
plicate  the  cube;  or  in  the  same  way  obtain  the  edge  of  a  cube  of  any  given 
number  of  times  the  volume  of  a  given  cube.  (The  pupil  may  show  that 
I~K3  =  2KB3 ;  also  that  AK3  =  2n<3.) 



THE  CONCHOID  OF  NICOMEDES. 

83.  DEF. — The  Conchoid  of  Nicomedes  is  the  locus  of  a 
point  in  a  line  which  revolves  on  and  slides  in  a  fixed  pivot,  so  as  to 
allow  a  constant  portion  of  the  line  to  project  beyond  a  fixed  right  line. 

84.  JProb.     To  construct  the  Conchoid  of  Nicomedes. 

SOLUTION.  —Let  O  be  the 
fixed  point,  or  pivot,  X'X 
the  fixed  line,  and  A  B  the 
constant  portion  of  the  re 
volving  line.  Draw  a  con 
venient  number  of  radiating 
lines  through  O,  and  on  each 
lay  off  above  X'X  the  dis 
tances  Cl,  FP,  E6,  etc., 
equal  to  A  B.  Then  wiU  1, 
2,  3,  4,  etc. ,  be  points  in  the 
locus  ;  and  M  B  N  will  be  the  conchoid. 

Sen.— This  locus  is  readily  drawn  by  mechanical  means.  Let  X'X  and 
YY'  be  two  bars  fixed  at  right  angles  to  each  other.  Let  any  one  of  the 
radiant  lines,  as  OP,  represent  a  ruler,  grooved  on  the  under  side  so  as  to 
slide  on  the  head  of  a  pin  fixed  in  the  bar  YY',  at  O.  Let  there  be  a 
fixed  pin  on  the  under  side  of  the  ruler,  as  at  F,  which  can  slide  in  a 
groove  on  the  upper  side  of  the  bar  X'X.  Now,  placing  the  groove  in  the 
ruler  on  the  head  of  the  pin  at  O,  and  the  pin  in  the  ruler,  in  the  groove  in 
X'X,  any  point  in  the  ruler,  as  P,  will  describe  the  conchoid. 


83.  Prob.     To  produce  the  equation  of  the  Conchoid  of  Nicomedes. 

SOT.TTTION.  —  Let  P,  Fig.  62,  be  any  point  in  the  locus  referred  to  the  axes 
XX',  Y  Y';  and  let  its  co-ordinates  A  E  and  PE,  be  a-,  and  ?/.  Let  A  B  =  a,  and 
AO  =  b.  Produce  PE  till  it  meets  O  D  drawn  parallel  to  AX.  Now,  by  simi* 
lar  triangles,  PE  :PD  ::EF  :  O  D  ;  or  y  :  y  -f  6  :  :  vV  —  y*  :  x.  Squaring, 
y2  :  (y  -f  &)2  :  :  a*  —  y*  :  «*.  .  •  .  x*y*  =  (y  4.  b^(ai  _  ^ 


-  -(-id  --),  for  every  positive  value  of 


Sen.  1.—  Since  x  =  ±  v/«2 

numerically  less  than  a,  x  has  two  numerically  equal  values  with  opposite 
signs  ;   which  values  increase  as  y  diminishes,  and  for  y  =  0,  x  =  db  oo. 


EQUATIONS  OF  HIGHER  PLANE  CURVES. 


Gl 


.  • .  This  portion  of  the  locus  is  symmetrical  with  respect  to  the  axis  of  y, 
and  has  the  axis  of  x  for  a  common  asymptote  of  its  two  branches.  Again, 
as  all  negative  values  of  y,  not  numerically  greater  than  a,  give  numerically 
equal  values  of  x  with  opposite  signs,  there  is  a  portion  of  the  locus  below 
the  axis  of  x,  which  is  also  symmetrical  with  respect  to  the  axis  of  y.  To 
discover  the  form  of  this  portion,  1st  consider  a  >•  b.  Then  f or  y  =  —  a, 
or  — b,  x  =  0,  but  for  values  of  y  between  these  two  limits,  x  has  two  nu 
merically  equal  values  with  opposite  signs  ;  hence  the  locus  between  these 
two  limits  is  an  oval  symmetrical  with  respect  to  the  axis  of  y.  For  y  nu 
merically  less  than  b,  and  negative,  the  values  of  x  increase  numerically  till, 
at  y  =  0,  they  become  ±  oo  ;  hence  between  O  and  the  axis  of  abscissas 
there  are  two  infinite  branches,  symmetrical  with  respect  to  the  axis  of  y, 
and  having  the  axis  of  x  as  a  common  asymptote.  2nd.  When  a  —  b  the 
oval  disappears.  These  forms  are 
described  mechanically  by  taking  the 
point  on  the  moving  ruler  below  the 
fixed  line. 


FIG.  63. 


FIG.  64. 


Sen.  2. — When  5  =  0,  the  equation 
becomes  .r2?/2  =  y2(a2  —  y2) ;  or  x2  +  3/2 
=  a2.  This  is  the  equation  of  the 
circle,  as  it  evidently  should  be. 

Sen.  3. — This  curve  was  invented 
by  the  geometer  whose  name  it  bears, 
for  a  purpose  similar  to  that  subserved 
by  the  cissoid.  The  problem  of  the 
Duplication  of  the  cube  and  the  Tri- 
section  of  an  angle  had  been  shown  to 
be  identical,  as  both  depend  upon  the 
insertion  of  two  means  in  a  continued 
proportion  between  two  extremes. 
Thus,  letting  a  and  b  be  the  extremes, 
it  is  required  to  find  x  and  y,  so  that 
a  :  x  :  y  :  b  ;  i.  e.,  a  :  x  :  :  x  :  y,  and 
x  :  y  : :  y  :  b.  This  problem,  viz. ,  the 
insertion  of  two  means  between  two 
extremes,  is  effected  by  the  cissoid. 
In  the  cissoid,  Fig.  59,  ED  and  AD' are 

two  means  between  AD  and  ID'  :  that  is,  AD  :  ED  :  AD'  :  I  D'. 
The  Trisection  of  an  angle  by  means  of  the  conchoid  is  effected  thus  :  Let 
COM,  Fig.  66,  be  the  angle  to  be  trisected.  From  any  point,  D,  in  one 
leg  let  fall  a  perpendicular,  DB,  on  the  other.  Take  CB  =  2 DO,  and 
with  O  as  the  fixed  point,  XX  as  the  fixed  line,  and  CO  as  the  ruler  with 
the  constant  portion  CB  projecting  beyond  X'X,  construct  the  arc  CR  of 
the  conchoid.  Erect  DP  perpendicular  to  X'X,  and  draw  PO.  Then  is 
POC  one- third  of  COM.  To  prove  this,  bisect  PH  as  at  E,  and  draw 


O 

FIG.  65. 


C2 


THE   CARTESIAN   METHOD    OF   CO-ORDINATES. 


DE.  Draw  also  FE  parallel  to  DH.  Since 
PE  =  EH,  PF=FD,  andED  =  PE  =  EH  = 
DO.  By  reason  of  the  isosceles  triangles  RED, 
and  EDO,  we  have  angle  DEO  =  2R  =2POC. 
But  DEO  =  EOD.  .-.  2EOC  =  EOD,  or 
EOC 


[NOTE.  —  This  scholium  is  by  no  means  necessary  to  the  in 
tegrity  of  the  course.  It  is  inserted  merely  as  a  matter  of 
interest  to  the  student,  giving  him  a  few  hints  upon  a  subject 
which  has  figured  so  prominently  in  the  history  of  geometry. 
It  will  afford  a  good  exercise  for  the  student  who  has  time  FlG.  66. 

and  ability,  to  demonstrate  fully  the  facts  hinted  at,  and  which 

are  not  demonstrated  above.  Thus,  let  him  show  why,  in  the  cissoid,  AD:ED:AD':ID';  also 
how  the  insertion  of  two  means  enables  us  to  obtain  any  multiple  of  the  cube  ;  also  how  the 
conchoid  effects  the  same  purpose  ;  and  that  the  two  problems  are  in  reality  but  one.] 


THE  WITCH  OF  AGNESL 

86.  DBF. — The  Witch  of  Agnesi  is  the  locus  of  the  extrem 
ity  of  an  ordinate  to  a  circle,  produced  until  the  produced  ordinate 
is  to  the  ordinate  itself,  as  the  diameter  of  a  circle  is  to  one  of  the 
segments  into  which  the  ordinate  divides  the  diameter, — these  seg 
ments  being  all  taken  on  the  same  side. 


87 •     J?rob.     To  construct  the  Witch  of  Agnesi. 

SOLUTION. — Let    AB    be    the    circle. 
Draw  a  series   of  parallel  ordinates  1  1, 
2  2,  P'P,  etc.      To  find  a 
point  in  the  locus,  take  P  E 
:  FE  :  :AB  :AE,   and  P 
is   such   a    point.     In  like 
manner  locate  other  points, 
as  1,  2,  etc. 


88.     I* rob.     To  produce  the  equation  of  the  Witch. 

SOLUTION. — Letting  the  axes  be  as  represented  in  Fig.  67,  so  that  P  being  any 
point,  A  D  =  a,  P  D  =  y,  and  calling  the  diameter  of  the  circle,  A  B  =  2a,  the 
equation  is  x"y  =  4a-(2a  —  y).  [Let  the  student  supply  the  demonstration.  ] 

Sen. — The  Witch  has  but  one  portion,  as  represented  in  the  figure  ;  it  is 
symmetrical  with  respect  to  the  axis  Y  Y',  is  comprised  between  ?/  =0,  and 
y  =  2a,  and  has  X'X  for  an  asymptote.  [Let  the  student  give  the  proof.] 


EQUATIONS  OF  HIGHER  PLANE  CURVES. 


63 


THE  LEMMSCATE  OF  BERNOUILLI. 

SO.  DEF. — The  Lemniscate  of  Hernouilli  is  a  curve  such 
that  the  product  of  two  lines  drawn  from  any  point  in  it  to  two  fixed 
points,  called  the  foci,  is  equal  to  the  square  of  half  the  distance  be 
tween  these  foci. 


00.  JProb.  To  construct  the  Lemniscate  of  Bernouilli. 

SOLUTION. — Let  F  andF'  be  the  foci.     From 
F'  as  a  centre,  with  any  convenient  radius,  as 
F'  P ,  draw  an  arc,  as  PG.    Find  a  third  propor 
tional  to  F'  P  and  F'  A.  Let  this  be  P  F.  From 
F  as  a  centre  with  this   third  proportional, 
draw  an  arc  intersecting  the  former  in  P  and 
6.     Then  will   P  and  6  be  points  in   the   locus  ;  for,  by 
construction    F'P  X   PF  =  AF2.     In  like   manner  find 
other  points.     Fig.  69,  shows  a  convenient  method  of  find 
ing  these   proportionals.     G  H  —  F'  F,  and  TG  —  A  F. 
Since    TL  :  TG  :  :  TG   :  TI,TL:AF::AF:TI  ; 
and  T  L  and  T I   are   the  corresponding  radii  to  be  used 
in  locating  a  point,  as  P.     With  one  pair  of  distances  four 
points  can  be  found. 


FIG. 


01.  Prob.  To  produce  the  equation  of  the  Lemniscate. 

SOLUTION.—  Assuming  XlX'  and  Y  Y'  as  axes  of  reference,  letting  P  be  any 
point  whose  co-ordinates  AD  and  PD,  are  x  and  y,  and  putting  AF  =  A  F' 
=  c,  we  have  the  distance  between  the  two  points  F'  and  P,  or  F  P 
=  \/(x  -\-  cf  -j-  y*.  In  like  manner  F  P  —  \/(c  —  x)*  +  y*.  Whence  by  definition 
2  X  \/(c—  z)24-  y2  =  c*,  or  (y*  -f  &)*=  2c-(x2  —  y2). 


-  y2  =  c*,  or  (y*  - 

Sen.  1.  —  Let  the  student  observe  the  symmetry  and  limits  of  the  curve 
from  its  equation.  Observe  that  in  the  construction  F'B  =  FC  —  TW. 
As  FB  X  F'B  =cs,  and  FB  =  AB  —  c,  and  F'B  —  AB  +  c,  we  find  that  AB 
—  Cv/2:  Putting  AB  =  a  =  c  v/2,  2c*  =  «2,  whence  the  equation  of  the  curve 
in  terms  of  its  semi-axis  is  (y2  -4-  #2)2  =  a?(x2  —  y2). 

*  Sen.  2.  —  (To  be  read  on  review.)  The  equation  of  the  Equilateral  Hy 
perbola  whose  semi-axis  is  a,  and  co-ordinates  x'  ,  y',  is  x'-  —  y'2  —  a'2.  The 
equation  of  its  tangent  is  xx'  —  yy'  =  a2.  The  equation  of  a  perpendicular 

from  the  centre  upon  the  tangent  is  x  =  --  -y.     From  these  three  equations, 
eliminating  x  and  y1  ',  that  is  finding  the  locus  of  the  intersection  of  a  per- 


64  THE    CARTESIAN   METHOD   OF   CO-OHDINATES. 

pendicular  from  the  centre  upon  the  tangent,  we  find  (.«*  +  3/':)2  =  a*(x°  —  #-)• 
Therefore  this  lemniscate  is  the  locus  of  the  intersection  of  a  perpendicular 
from  the  centre  of  an  equilateral  hyperbola  upon  its  tangent,  the  axes  of 
both  loci  being  coincident. 


THE  CYCLOID. 

92.  DEF. — The  Cycloid  is  the  locus  of  a  point  in  the  circum 
ference  of  a  circle  which  rolls  along  a  fixed  right  line. 


Sen. — The  cycloid  can  be  constructed 
mechanically  by  rolling  a  wheel,  as 
HPI,  Fig.  70,  along  the  edge  of  a 
fixed  ruler,  as  AX.  A  point  P  in  the 
circumference  of  the  wheel  describes 
the  cycloid. 


FIG.  70. 


03.  DBF'S. — The  circle  H  P I  is  called  the  Generating  Circle, 
or,  simply  the  Generatrix  ;  A  X  is  the  IZase,  and  is  equal  to  the 
circumference  of  the  generatrix ;  and  B  F,  erected  perpendicular  to 
the  base  at  its  centre,  is  the  Amis,  and  is  equal  to  the  diameter  of 
the  generatrix. 


04.  IProl). — Having  given  the  cycloid,  to  put  the  generating  circle  in 
position. 

SOLUTION. — There  is  given  simply  the  curve  A  BX,  Fiq.  70.  Draw  the  base 
AX,  and  bisect  it  by  the  perpendicular  BF.  BF  is  the  axis.  Bisect  the  axis 
by  N  K  drawn  parallel  to  the  base.  Now,  to  put  the  generating  circle  in  the  posi 
tion  it  occupied  when  the  generating  point  was  at  P.  draw  from  P  as  a  centre, 
with  a  radius  equal  to  the  radius  of  the  generatrix  (BO  or  OF),  an  arc  cutting 
N  K,  as  at  C.  C  is  the  centre  of  the  generatrix. 


05.  frob. — To  produce  the  equation  of  the  cycloid  referred  to  its 
base  and  a  perpendicular  at  the  left  hand  vertex. 

SOLUTION.— Let  P,  Fig.  70,  be  any  point  in  the  cycloid  A  BX,  referred  to  AY, 
and  A  X  as  axes.  Then  A  D  =  x,  and  P  D  =  y.  Call  the  radius  of  the  generatrix 
r.  Now  A  D  =  A I  —  D  I .  But  by  construction,  A I  =  arc  P I  =  versin  -  ]  I  L, 
or  versin-  ly,  to  a  radius  r.  Dl  =  PL  =  \/TL  X  LH  =  \/'y(2r  —  y) 
=  \/'2ry  —  y2.  .  • .  x  =  versin— '  y  —  \/2ry  —  y*. 

SCH. — If  y  be  negative,  v2ry  —  y*  becomes  imaginary  ;  hence  the  curve 
lies  on  but  one  side  of  the  base.  For  y  =  0,  we  have  x  =  versin  ~ l  0  =  0, 


EQUATIONS  OF  HIGHER  PLANE  CURVES.  65 

27rr,  4:7tr,  etc.,  etc.  Hence  we  see  that  there  are  an  infinite  number  of  arcs 
like  ABX,  belonging  to  the  curve.  This  is  also  apparent  from  the  defini 
tion,  as  each  revolution  of  the  generatrix  produces  one  arc,  and  there  is  no 
limit  to  the  number  of  revolutions.  For  y  =  2r,  x  =  versin  ~ l  (2r)  =  TTT, 
Sxr,  5jtr,  etc.,  etc.,  as  it  should  from  the  construction. 


96.  I* rob. — To  produce  the  equation  of  the  cycloid  referred  to  its 
axis  as  the  axis  of  abscissas,  and  a  tangent  at  the  vertex  of  the  axis  (  B,  Fig. 
71),  as  the  axis  of  ordinates. 

SOLUTION.— Let  PM  =y,  and  BM=iC.  Now  PM=PL-fLM.  But 
PL=\/2ra: — xli  an(i  L-M  =  IF  =  AF  —  Al=the  semi-circumference  of 
the  generatrix  —  arc  PI.  Again,  arc  PI  =  versin-1  (2r  —  x).  .'.  y  = 

V/^/'X  —  tf  -f  rtr —  versin -'  (2r  —  x). 

97.  COR.    PR  =  arc    B  R,   P   be 
ing  any  point  in  the  curve. 

For    PR  =  L  M   =  A  F  —  A I   =  arc 
HPI  —  arc  PI  =arc  HP  =  arc  BR.  A        ^ 

FIG.  71. 

gCHi  1. — Considering  the  equation  x  =  versin-'y  —  \/2ry  —  y'*,  we 
observe  that  there  are  an  infinite  number  of  values  of  x  for  every  value  of 
y.  First  of  all,  the  term  versin-1  is  ambiguous  as  to  its  sign,  since  a  nega 
tive  arc  has  the  same  versed-sine  as  the  numerically  equal  positive  arc. 
Moreover,  whatever  a  versed-sine  may  be,  there  are  not  only  the  -f  and  — 
arcs  less  than  180°,  and  also  the  -f-  an^  —  arcs  °f  360°  —  the  former,  which 
corresponds  to  it,  but  these  increased  numerically  by  every  multiple  of  2  jr. 
We  are  therefore  to  write  the  term  versin~~ 'y  with  the  sign  d=,  and  under 
stand  that  for  every  value  of  y  it  has  an  infinite  number  of  numerical  values, 
each  succeeding  value  in  the  series,  being  numerically  2#  greater  than  the 
preceding.  In  the  second  place  the  term  —  \/%ry  —  y\  being  a  square 
root  is  to  be  written  —  (± v/2ry  —  y2),  or  qp \/2ry  —  y2.  Writing  the  equa 
tion  in  this  way  we  have  x  =  ±  versin~]y  ^  y%ry  —  y-.  The  geometrical 
significance  of  these  facts  is  as  follows.  Let  y  have  any  value  as  PD, 
Fig.  70  ;  then  x  has  1st,  the  positive  value  AD  and  gives  the  point  P,  and 
a  numerically  equal  negative  value,  giving  a  point  P'  similarly  situated  on 
the  left  of  AY,  if  we  take  versin-1  y  <  180° ;  but  if  we  take  360°  —  this 
arc,  both  -f-  and  —  as  the  value  of  versin~'y,  we  get  two  other  values  of  x, 
one  -[-  and  the  other  — .  The  former  is  where  PM,  Fig.  70,  produced  to 
the  right  meets  the  curve,  and  the  other  the  corresponding  point  on  the 
left  of  AY.  Taking  for  values  of  versin-!y,  the  values  now  considered 
+  27f  simply  repeats  the  curves  at  a  distance  2;r  both  at  the  right  and  left. 
The  equation  in  (96)  has  a  similar  interpretation. 


THE   CARTESIAN   METHOD   OF   CO-ORDINATES. 


SCH.  2. — The  Cycloid  is  a  transcendental  curve,  and  is  next  to  the  Conic 
Sections  in  importance  among  plane  loci. 

OS.  It  frequently  occurs  that  the  equation  of  a  locus  can  be  writ 
ten  immediately  from  the  definition.  The  Sinusoid  is  of  this  char 
acter.  The  definition  is,  "  The  Sinusoid  is  the  locus  of  a  point  whose 
abscissa  is  the  arc,  while  its  ordinate  is  the  sine  of  the  arc."  Hence 
the  equation  is  y  =  sin  x.  The  other  simple  trigonometrical  curves 
(page  16,  Ex's  27 — 33)  are  of  the  same  character,  as  is  also  the  loga 
rithmic  curve,  x  =  log  y. 

00.  As  has  been  remarked  before,  the  number  of  plane  curves  is 
infinite.  The  foregoing  have  been  given  as  specimens,  from  which  it 
is  hoped  that  the  student  will  be  able  to  learn  how  the  equations  of 
loci  referred  to  rectangular  axes  are  produced  from  the  definitions  of 
the  loci.  A  great  many  kinds  of  curves  are  suggested  by  tne  study 
of  the  Properties  of  Carves.  Some  of  these 
will  be  noticed  hereafter.  Again,  Mechan- 
ics  and  other  branches  of  Physics,  give  rise 
to  the  study  of  curves,  the  production  of 
whose  equations  requires  a  knowledge  of 
the  principles  of  Natural  Philosophy. 
Thus,  the  Tractrix  is  the  path  described  by  a 


A' 

FIG.   72. 

weight,  W,  Fig.  72,  to  which  a  cord,  AW,  is  attached,  and  the  extrem 
ity,  A,  of  the  cord,  made  to  pass  over  the  path,  AB,  friction  being- 
supposed  uniform  and  perfect.  Again,  the  Catenary  is  the  line  which 
a  perfectly  flexible  chain  assumes  when  its  ends  are  fastened  at  two 
points,  as  A  and  B ,  Fig.  73,  nearer  together 
than  the  length  of  the  chain.  Caustics  are  an 
interesting  class  of  curves  consequent  upon  the 
laws  of  reflected  light.  The  next  chapter  gives 
several  varieties  of  curves  called  Spirals. 


FIG.  73. 


CHAPTER  IL 

THE  METHOD  OF  POLAR  CO-ORDINATES. 


SECTION  I. 
Of  the  Point  in  a  Plane, 

1 00.  Prop. —  Tfie  Position  of  any  Point  in  a  plane  can  be  designated 
by  giving  its  Distance  and  Direction  from  a  fixed  point  in  the  plane.     In 
order  to  indicate  direction,  a  fixed  line  has  to  be  assumed. 

ILL. — Let  A,  Fig.  74,  be  the  fixed  point,  and 
AX  the  fixed  line.  Let  r  represent  the  distance 
from  the  fixed  point  to  the  point  to  be  desig 
nated,  as  AP,  AP',  etc.,  and  0  the  angle  in 
cluded  between  the  fixed  line  and  the  line  from 
the  fixed  point  to  the  point  to  be  designated,  as 
PAX,  P'  AX,  etc.,  etc.  It  is  evident  that  by  FIG.  74. 

giving  all  possible  values  to  0,  and  r,  all  points  in  the  plane  of  the  paper  may  be 
located.  Thus,  for  P,  we  have  0  =35°,  r  =  5  ;  for  P',  0  =  120°,  r  =  10  ;  P'', 
0=195°,r=8  ;  and  for  P'",  0=345°,  r  =  ll. 

101.  DEF'S. — TheJPole  is  the  assumed  fixed  point,  as  A.     The 
Prime  Radius  (called  also  the  Initial  Line,  and  the  Polar  Axis} 
is  the  assumed  fixed  line,  as  AX.     The  Radius  vector  is  the 

distance  from  the  pole  to  the  point  to  be  designated,  as  A  P,  A  P', 
etc.  The  Variable  Angle  (0)  is  the  angle  indicating  the  di 
rection  of  the  point  from  the  pole.  When  0  is  reckoned  around  from 
right  to  left,  it  is  called  +  ;  when  reckoned  from  left  to  right,  — . 
The  radius  is  -f  when  estimated  in  the  direction  of  the  extremity  of 
the  arc  measuring  the  variable  angle ;  and  it  is  —  when  estimated  in 
the  opposite  direction,  r  and  0  are  the  Polar  Co-ordinates. 


102.  Prop. — The  Polar  equations  of  a  Point  are,  r  =  a,  and  0  =  6  ; 

since,  by  giving  suitable  values  to  r  and  0,  all  points  in  the  plane  can  be 
located. 

Ex.  1.  Locate  r  =  5,  0  =  \it. 

SOLUTION. — The  radius  being  1,  it  is  the  semi-circumference.  Hence  %it  =  60°. 
Now,  lay  off  PAX  =  0  =  60°,  Fig.  74  ;  and,  taking  AP  =  5,  P  is  the  required 
point. 


68  THE  METHOD   OF  POLAB   CO-OKDINATES. 

Exs.  2  to  6.  Locate  r  =  3,  0  =  %7e:  r  =  4,  0  =  %7r:  r  =  6,  0=*7t : 
r  =  —6,  0  =  135°  :  r  =  4,  0  =  —60°. 

Ex.  7.  Show  that  r  =  10,  0  =  — 45°,  is  the  same  point  as  r  =  —10, 
0=135°. 


103.  J*rob. — To  find  the  Distance  between  two  points  given  by  their 
polar  co-ordinates. 

SOLUTION. —Let  the  co-ordinates  of  P,   Fig.  75,  be  P 

(r,  6)  ;  and  of  P',  (r',  0').     We  are  to  find  PP'  in 
terms    of   r   r',   0,    and    6'.      Now,   in    the    triangle 

PAP',  AP  =  r,  AP'  =r'and  the  included  angle    „ 

PAP'  =  0  —  0'.     Hence,  representing  PP'  by  D,  we   A  X. 

have  from  principles  of  trigonometry  FIG.  75. 

D  =  \/r*  -f-  r'a  —  2rr'  cos  (8  —  0~),     Q.  E.  D. 

Ex.  1.  Find  the  distance  between  r  =  3,  0  =  ^7?,  and  r  —  4,  0 4?r. 

Ex.  2.  Find  the  distance  between  (8,  f  TT),  and  (3,  $4yr). 

Ex.  3.  Find  the  distance  between  (v/2,  45°),  and  (1,  0°). 

Results  in  the  last  three  examples,  not  in  order,  7,  1,  5. 


SECTION  II. 
Of  the  Eight  Line, 

104.  Prob.—To  produce  the  Polar  Equation  of  the  Right  Line. 

SOLUTION.—  The  form  of  this  equation  (like  all  others)  depends  upon  the  con 
stant*  assumed.     We  will  consider  two  forms. 

^  1st.  When  the  constants  are  the  length  of  the  perpendicular  from  the  pole  upon  the 
line,  and  the  angle  which  this  perpendicular  makes  with  the  prime  radius.  Thus  in 
Fig.  76,  let  M  N  be  any  line  ;  A,  the  pole  ;  AX,  the 
prime  radius  ;  the  perpendicular  from  the  pole  upon  the 
line,  A  D  =  p  ;  and  the  angle  which  the  perpendicular 
makes  with  the  prime  radius,  DAX  =  a.  Let  P  be 
any  point  in  the  line  MN,  and  its  co-ordinates  be 
(r,  0).  Now,  in  the  right  angled  triangle  PAD,  we 
have  AD  =  AP  cos  PAD,  or  p  =  r  cos  (0  —  a); 

\N 


2nd.    When  the  constants  are  the  intercept  on  the  prime  FlG-  76. 


OF  THE   BIGHT  LINE.  •    69 

radius,  and  the  angle  which  the  line  makes  with  the 

prime  radius.     In  Fig.  77,  let   M  N   be  any  line 

referred  to  the  pole,  A,  and  the  prime  radius,  AX. 

Represent  the  intercept,  A~T,  by  c,  and  the  angle 

NTX,  by  a.     Let   P  be  any  point  in  the  line, 

and  its    co-ordinates   AP  =  r,   and  PAX  =  0. 

The  angle  "TPA  =  6  —  a  ;  and,  from  the  triangle 

PTA,wehaveAP:AT::sinPTA:sinTPA,  FIG.  77- 

or  r  :  c  : :  sin  a  :  sin  (0  —  a)  ; 

sin  a 

.  • .   r  =  —. — r c.     Q.  E.  D. 

sin  (Q  —  a) 

SCH.  1.  — Discussion  of  the  equation  r  = = .     When  0  =  0,  we  have 

cos  (0  —  a) 

r  = .     This  is  as  it  should  be,  for,  when  0  =  0,  r  =  AT,  Fiq.  76, 

cos  ( —  a) 

which,  from  the  triangle  ADT,  is  seen  to  be  — ,  or —  .     The 

cos  DAT          cos( — a) 

—  sign  of  a  indicates  that  the  radius  vector  falls  upon  the  opposite  side  of 
the  perpendicular  from  that  assumed  in  producing  the  equation.  .  .  .  When 

0  =  a,   r  =  -2L.  =  jp,  as  it  evidently  should.  .  .  .  When  0  —  a  =  90°, 
cosO 

P 
r  =  -  =  oo.     In  this  case  the  radius  vector  becomes  parallel  to  the  line, 

and  hence  oo .  .  .  .  From  0  —  a  =  90°  to  0  —  a.  =  270°,  r  is  negative,  as 
it  should  be ;  since,  in  order  to  reach  the  line  M  N ,  it  must  be  produced 
backward^  i.  e. ,  from  the  pole  in  a  direction  opposite  to  the  extremity  of  the 
arc  0  measured  from  the  prime  radius  around  to  the  right.  .  .  .  From  0  — 
a  =  270°  to  0  —  a  =  360°,  r  is  -f  ;  and  at  0  —  a  =  360°,  r  =  p,  as  it  should. 

Also,  at  0  =  360°,  r  = ^ =  AT.  ...  When  the  line  M  N  passes 

cos( — a) 

through  the  pole,  r  =  ,  which  is  0  for  all  values  of  0  except  0  = 

cos  (0  —  a) 

(90°  -f  «),  for  which  r  =  -,  i.  e.,  indeterminate.  The  0  values  of  r  indi 
cate  that  we  have  not  to  pass  any  distance  from  the  pole  to  reach  the  line  ; 
and  the  -  value  indicates  that  for  all  values  of  r,  its  extremity  is  in  the  line 
MM-  .  .  •  Finally,  when  a  =  0,  the  line  MN  becomes  perpendicular  to  the 

prime  radius,  and  its  equation  is  r  =  — — • 

cos  0 

sin  oc. 
SCH.  2. — Discussion  of  the  form  r  = c.     For  0  =  0,  r  =  — c  ; 

sin  (0  —  a) 

which  is  evidently  correct,  as  it  is  reckoned  backward,  and  equals  c,  in 
length.  .  .  .  For  all  values  of  0  <<  a,  r  is  negative,  and  hence  is  reckoned  back- 


70  THE    METHOD   OF    POLAK    CO-OEDINATES. 

ward.  .  .  .  For  0  =  a,  r  =  c  =  oo,  as  it  should,  since  it  is  then  par 
allel  to  the  line.  .  .  .  For  values  between  6  >•  «,  and  9  =  180°  +  a,  r  is 
positive.  ...  At  0  =  180°  -f  a,  r  becomes  infinite  ;  and,  when  0  passes 

sin  (X 

180°  +  a,  r  is  again  negative.  .  .  .  For  0  =  180°,  r  =    .  c  =  c, 

sin  (180°  —  a.) 

i.  e.,  AT.  .  .  .  If  the  line  MN  passes  through  the  pole,  c  =  0,  whence 

r  = =  =  0,  for  all  values  of  0  except  0  =  a,  in  which 

sin  (0  —  a)       sin  (0  —  a) 

case  r  =  -.     These  results  are  evidently  correct,  for  in  the  former  cases  we 

have  to  go  0  distance  from  the  pole  in  the  specified  directions,  in  order  to 
reach  the  line  ;  and,  in  the  latter  (when  0  =  a)  the  radius  vector  falling  on 
the  line  MN,  its  extremity  is  equally  in  the  line  for  all  values  of  r. 

Ex.  1.  What  is  the  polar  equation  (first  form)  of  a  line  the  nearest 
point  in  which  is  6  from  the  pole,  and  the  perpendicular  to  which 
makes  an  angle  of  45°  with  the  prime  radius?  Where  does  this  line 
cut  the  prime  radius  ?  For  what  values  of  0  is  r  infinite  ?  What  is 
the  value  of  r  for  0  =  75°  ?  For  0  =  15°  ?  Construct  these  values 
of  r  and  verify  them  by  drawing  the  line. 

Ex.  2.  What  is  the  polar  equation  (first  form)  of  a  right  line  per 
pendicular  to  the  prime  radius,  and  which  cuts  it  at  4  to  the  left  of 
the  pole?  What  is  the  value  of  r  when  0=  60°  ?  Why  is  the  sign 
of  r  negative  in  the  latter  case  ?  Between  what  values  of  0  is  r  posi 
tive  ?  What  is  the  value  of  r  when  0  is  120°  ? 

4 
The  equation  is  r  = . 

COS0 

Ex.  3.  Give  the  equation  of  a  line  parallel  to  the  prime  radius,  and 


10  above  it ;  also  at  m  below.       The  latter  equation  is,  r  = : 


111 


sin  & 


SECTION  III. 
Of  the  Circle, 

f  rob.— To  produce  the  Polar  Equation  of  a  Circle,  the  pole 
being  in  the  circumference,  and  the  polar  axis  being  a  diameter. 


OF  THE   CIRCLE. 

SOLUTION.— Let  A,  Fig.  78,  be  the  pole  ;  E, 
the  radius  of  the  circle  ;  and  P,  any  point  in  the 
circumference.  Then  AP  =  r,  and  P  A  B  =  0. 
Now,  from  the  right  angled  triangle  APB,  \ve 
have,  by  trigonometry,  r  =  2R  cos  0.  Q.  E.  D. 

Sen.  1. — Discussion  of  the  Equation.  If  0 
=  0,  r  =  2R.  If  O  =  X  or  f  tf,  r  =  0.  For  FIG.  78. 

values  of  0  between  \Tt  and  §#,  r  is  negative,  indicating  that  for  these  val 
ues  of  0  the  radius  vector  must  be  produced  backward  to  meet  the  circum 
ference.  (The  student  should  observe  that  the  results  obtained  from  the 
equation,  accord  with  the  values  as  observed  from  the  figure.) 


>  2. — If  the  pole  is  at  the  centre,  the  equation  is  evidently  r  =  E,  for 
all  values  of  Q. 


10  G.  J?TOb*  —  To  produce  the  General  Polar  Equation  of  a  Circle. 

SOLUTION.—  Let  the  constants  be  (r,  B'),  the 
co-ordinates  of  the  centre,  C,  Fig.  79  ;  and  E, 
the  radius  of  the  circle.  Let  P  be  any  point  in 
the  circumference,  and  its  co-ordinates,  A  P  and 
the  angle  P  AX  ,  be  r  and  0.  Now,  since  the  dis 
tance  between  the  points  C  and  P  is  constant  (R), 
we  have  from  (lO3)R=\/r*+r'*—  2rr'cos(0—  6'j.  A 
Whence  we  have,  FIG.  79. 

r«  —  2r'  cos  (0  —  0'  )r  =  J2a  —  r'2.     Q.  E.  D. 

Sen.  —  Discussion  of  the  Equation.  —  Solving  the  equation  for  r,  we  nave, 
r  =  r'  cos  (0  —  0')  rfc  \/2*  —  r'*  sina  (0  —  0').  This  value  of  r  is  real  only  for 
such  values  of  0  as  render  r'2  sin2(0  —  0')  <or  =  jR*  ;  i.  e.,  when  sin(0  —  Q') 

is  numerically  <  or  =  —  .    Now,  sin  (0  —  0')  =  sin  P  AC  =  —  =  -        makes 


R* 

P  a  right  angle,  and  hence,  A  P  tangent  to  the  circle.    But  sin2(0  —  6  ')  =  —  - 

D 

gives  sin  (0  —  0')  =  i  —  ;  hence  there  are  /too  positions  of  r  in  which  it  is 

tangent  to  the  circle.     (This  is  the  familiar  truth  of  Geometry,  that  from 
any  point  without  a  circle  two  tangents  can  be  drawn  to  the  curve.)     The 

•ry 

condition  sin  (0  —  0')  =  --  r  is  satisfied  by  the  lower  point  of  tangency,  for 
•which  0  —  0'  becomes  —  ,  and  hence,  sin  (0  —  0')  is  —  .     Between  the  limits 

•n  T> 

gin  (0  —  0')  =  —  -  and  --  r,  there  are  two  real  and  unequal  values  of  r  for 


72  THE   METHOD   OF   POLAR   CO-ORDINATES. 

every  value  of  0,  as  evidently  should  be  the  case,  since  between  these  limits 
the  radius  vector  meets  the  curve  in  two  points  ----  When  VR*  —  r'*sui\ti  —  6'  ) 
<_»•'  cos  (0  —  0')  the  two  values  of  r  have  the  same  sign;  but  when 


*  —  r"~siir2  (6  —  0')  >•  r'  cos  (0  —  0'),  r  has  different  signs.  In  the 
former  case  the  pole  is  without  the  circle,  —  in  the  latter  within.  These 
facts  readily  appear  by  solving  the  inequality.  Thus,  in  the  latter, 
£2  —  r'9  Sin2(0  —  ©')>>  r'2  cos2(0  —  0'),  or  R*  >>  r'2  sin2(0  —  0')  +  r'2  cos*(0  —  0'). 
But  the  latter  member  reduces  to  r'2.  .'.  jR>r',  which  puts  the  pole 
within  the  circle.*  (For  other  cases,  see  examples  below.) 

Ex.  1.  What  is  the  equation  of  a  circle  whose  radius  is  10,  and  the 
polar  co-ordinates  of  whose  centre  are  (15,  J-TT)  ?  What  values  of  0  in 
dicate  that  the  radius  vector  is  tangent  to  the  circle  ?  Between  what 
limits  of  0  is  r  real  ?  Between  what  imaginary  ?  What  is  the  posi 
tion,  and  what  are  the  values  of  0  when  the  radius  vector  passes 
through  the  centre  ?  Construct  the  figure. 

The  equation  is  r2  —  30  sin  Or  =  —  125  ;  or  r  =  15  sin  0  ± 
\/100  —  225  cos5  0.  The  positions  of  tanyency  are  cos  0  =  -f, 
and  cos  0  =  —  |-,  when  r  =  5x/5. 

Ex.  2.  Give  and  discuss  as  above  the  polar  equation  of  the  circle 
whose  centre  is  at  (8,  JTT),  and  whose  radius  is  10.  Does  this  circle 
cut  the  polar  axis  ;  and,  if  so,  where  ?  How  do  you  determine  this 
point  from  the  equation*? 

Equation,  r2  —  8v/2  (sin  0  -f  cos  0)r  =  36.     Condition  of  tangency, 
sin  0  -f-  cos  0  =  -|  v7  —  2.     .'.  The  pole  is  within;  as  appears  also 

from  —  >  1.     Cuts  the  polar  axis  at  r  =  (4  db  v/34  )  v/2. 

Ex.  3.  Show  that  the  polar  equation  of  a  circle  is  r2  —  2r'r  cos  0 
=  R2  —  r'9  when  the  polar  axis  passes  through  the  centre  and  the  pole 
is  without  or  within  the  circle.  (Prove  this  directly  from  a  figure 
without  reference  to  the  preceding  forms.) 

Ex.  4.  Deduce  from  the  general  form  in  106,  the  form  in  105, 
and  also  the  one  in  Ex.  3. 


*  This  may  also  be  observed  from  sin  (Q  —  0',)  =  +  -,   which  is  the  condition    of   tangency. 

r 

This  is  possible  only  when  R  <  r,  t.  e.,  when  the  pole  is  without  the  circle.     When  R  =  r,  the  two 
tangents  become  one. 


OF   THE   CONIC   SECTIONS. 


73 


Ex.  5.  Let  the  student  show  from  the 
annexed  figure  that  in  the  equation  r 
=  r'cos  0_0'±  \/^--— 


the  rational  part,  r'  cos  (0  —  0'),  is  the 
chord,  A  D,  of  the  circle  described  up 
on  AC  (=O  as  a  diameter,  and  that 
the  radical  part,  ^  '  R*—  r'*  sin2  (0  —  0'), 
is  PD=P'D,  +  for  PD  and  —  for 
P'  D.  B  and  B',  are  the  points  where  the  radical  becomes  0. 


SUCTION  IV. 

Of  the  Conic  Sections, 

107.  I*rob. — To  produce  the  Polar  Equation  of  a  Conic  Section. 


SOLUTION.— Let  P  be  any  point  in  the  curve  ;  EC  the 
directrix  ;  A,  the  focus  and  pole  ;  and  AX,  the  axis  of 
the  curve  ana  the  polar  axis.  Let  2p  be  the  latus  rectum ;  e, 
Boscovich's  ratio,  whence  p=CAXe;  AP=r,  and 
PAX=0.  Then  AP  =  CD  X  e  =(CA  +  AD)e. 

But     C  A    =  —  ,   and    A  D  =  r  cos  0 ;  whence,  r  = 


IL_J_r  COB0V 


P 


1  —  e  cos  0 


Q. 


FIG.  80. 


108.  COR,  I.— The  Polar  Equation  of  the  JParabola. 

Since  in  the  parabola  e  =  1,  its  polar  equation  is 


1  —  Cos  0 


,  or  r  =  - 


versm  0 


W9.  COB.  2. — The  Polar  Equation  of  the  Ellipse  and 
Hyperbola  in  terms  of  the  semi-transverse  axis  and 
eccentricity.  Since  Boscovich's  ratio  (e)  and  the  eccentricity  are 
the  same  (48),  and,  numerically,  p  =  A(l  —  e2),  this  equation  is 


T  ~ — 


1  —  e  cos  6 
Sen.  1.— Discussion  of  the  Polar  Equation  of  the  Parabola.— For  0  =  0, 


74 


THE   METHOD   OF  POLAR   CO-ORDINATES. 


r  =  -  —  -  --  -  becomes  r  =?  —  —  —  oo  ;  i.  e.,  the  radius  vector  falling  upon 
1  —  cos  0  1  —  1 

the  axis  does  not  meet  the  curve  .....  For  any  value  of  0  >  0,  however  small, 
r  is  finite  ;  which  shows  that,  if  a  line  be  drawn  from  the  focus  making  any 
angle  however  small  with  the  axis  of  the  curve,  it  meets  the  curve  at  a  finite 
distance  .....  For  0  =  90°,  r  =  p,  as  it  should  .....  For  0  =  180°,  r  = 

-  --  —  =  -  -  =  \p.     This  is  evidently  correct,  since  r  becomes  A  B, 

Fig.  80,  when  0  =  180°.  But,  in  the  parabola,  A  B  =  %p  .....  For  0  =  270°, 
r  =  p,  as  it  should.  (Let  the  student  discuss  in  like  manner  the  form 


. 
versin  0 

SCH.   2.  —  Discussion    of    the    equation    r  = 

,        ~G  .  for   the  Ellipse.—  For  0  =  0,  r  = 
1  —  e  cos  0 

—  =  A(l  -\-e)  =A  4-  Ae;  which  makes 
1  —  e 

AC  =  A  -f  Ae,  as  it  should  (49)  .....  For  the 
point  P  at  the  extremity  of  the  conjugate  axis, 

Fig.  81,    cos  0   =   —  -  =  —  .      Hence   r  = 


-= 


AP 

or   r  -  Ae*  =  A(l 


<*). 


agreeably  to  (46}  .....     For  0  =  90°,  r  =  A(l  —  e*}  =  p  .....     For  0  = 

^4  n  _  e-2\ 
180°,  r  =  —  L_  -  1  —  A  —  Ae,  which  is  the  value  of  AB,  as  it  should 


be. 


SCH.  3.  —  Discussion  of  the  Equa- 


tton  r  =  -  --  -  for  the  Hyper- 
1  —  e  cos  0 

bola.  —  Kemembering  that  in  the 
hyperbola  e  >  1,  we  observe  that 
A  (1  —  e»)  is  essentially  negative,  and 
hence  that  the  sign  of  r  depends 
upon  the  sign  of  the  denominator, 
1  —  e  cos0  ;  r  being  -f  when  e  cos  0>1, 
and  —  when  e  cos  0  <  1.  Now  for 


:0,   r  = 


l—e 


A  +-  Ae, 


FIG.  82. 


which  indicates  that  A,  Fig.  82,  is  the  pole,  and  C  is  the  point  located 
when  0  =  0,  as  AC  =  A  -f  Ae  (49) As  0  increases  from  0,  cos  0  dimin 
ishes,  which  diminishes  the  numerical  value  of  1  —  e  cos  0,  till  0  =  cos-1- 

e 

(though  leaving  it  negative,  and  hence  r  positive),  hence  r  increases  and  the 


OF  THE   CONIC   SECTIONS.  75 

arc  C  M  is  traced When  1  —  e  cos  6  =  0,  i.  e. ,  when  cos  6 = -,( in  which  case 

e  \ 

0  =  cos"1-  Y  r  =  oo.    In  this  position  r  ( A  P ' )  becomes  parallel  to  the  asymptote 
<?/ 

OC       A       1 

OT,  a  line  so  drawn  that  DC  =  B,  whence  cosDOC  =  —  —  =  —  =  - 

OD      Ae      e 

When  0  passes  the  value  cos"1-,  1—  e  cos  0  becomes  positive  and  renders  r  nega 
tive,  and  the  left  hand  branch  begins  to  be  traced,  the  point  moving  in  the  di 
rection  M '  B  M  " .  Thus,  when  0  =  ZZA  X ,  r  being  negative  is  reckoned  back- 

_^|  n  e>2\ 

wards  and  the  point  P"  is  located.     When  0  =  90°,  r  —  — — -— —  =  —  p,  and 

P'"  is  located,  AP'"  being  equal  to  — p.     As  0  passes  from  90°  to  180°,  the 

A(l—e*}        A(l—e*) 
arc  P'" B  is  traced.     At  0  =  180°,  r  =    __   ,L  =     1        -  =  A  —  Ae  = 

—  A  B,  as  it  should.  (In  like  manner  let  fhe  student  observe  that  as  0  passes 
from  180°  to  270°,  BPIV  is  traced  ;  at  0  =  270°,  r  =  —  p  =  APIV  ;  from 

0  _  270°  to  0  =  cos"1-  in  the  Uh  quadrant,  r  remains  negative  and  P1VM"  is 

traced;  at  0  =  cos-1-  in  the  4th  quadrant  r  —  oo  =  APV,  parallel  to  the 

asymptote  OT'  ;  and  that,  finally,  from  this  value  of  0  to  0  =  360°,  r  be 
comes  positive  again,  and  the  arc  M  "  C  is  traced.) 

Ex.  1.  What  is  the  polar  equation  of  a  parabola  whose  principal 
parameter  is  8,  the  pole  being  at  the  focus  ?  What  is  the  length  of 

the  radius  vector  for  0  =  60°  ? 

4 

The  equation  is  r  = -.    For  0  =  60°,  r  =  8. 

1  —  cos  0 

Ex.  2.  "What  is  the  polar  equation  of  an  ellipse  whose  axes  are  12 
and  8,  the  pole  being  at  the  focus?  What  are  the  focal  distances  ; 
i.  e.,  what  are  the  respective  values  of  r  for  0  =  0,  and  for  0  =  180°  ? 
What  is  the  semi-lotus  rectum  ;  i.  e.,  what  is  the  value  of  r  when  0 

=  90°? 

g 

The  equation  is  r  = .     The  focal   distances   are   6  -f 

3_  v/5  cos  0 

2\/5,  and  6  — 2 v/5.     The  semi-lotus  rectum  is  2f . 
Ex.  3.  What  is  the  polar  equation  of  an  hyperbola  whose  trans 
verse  axis  is  6,  and  the  distance  between  the  foci  10  ?     What  is  the 
value  of  r  when  0  =  0?     When  0  =  90°?    When  0=180°?      What 

is  the  value  of  0  when  r  =  oo  ? 

i  r* 
The  equation  is  r  =      5  CQS  ^  _  3-     For  r===  °°»  e  =  53°8'>  nearlv- 


76  THE  METHOD   OF   POLAR   CO-ORDINATES. 

Ex.  4.  Show   that   the   polar  equation  of  the   parabola   become* 

ff\ 

r= — .  when  the  prime  radius  lies  in  the  position  AC,  Fiq. 

1  -f-  cos  0' 

80,  and  6'  is  reckoned  from  it  around  to  the  left  (in  the  usual  direc 
tion).     Show  that  in  this  form,  for  0'  =  0,  r  =  \p  ;  and  for  0  =  180°, 

7'=  00. 

BUG.— This  form  is  deduced  from  r  = ^  by  substituting  for  0,  9'— 180°, 

1  — -  COB  vj 

as  the  initial  line  is  revolved  forward  180°. 

Ex.  5.  A  comet  is  moving  in  a  parabolic  orbit  around  the  sun  at 
its  focus,  and  when  at  100,000,000  miles  from  the  sun,  the  radius  vec 
tor  makes  an  angle  with  the  axis  of  the  orbit  of  60°.  What  is  the 
polar  equation  of  the  comet's  orbit  ?  How  near  does  it  approach  the 
sun?  How  does  it  appear  in  a  parabola  that  r  =  2p,  for  0  =  60°  ? 

50,000,000 

The  equation  is  r  =  — . 

1  —  cos  0 


SECTION  V. 
Of  Higher  Plane  Curves, 


[TforE. — All  of  this  section,  except  the  portion  upon  Spirals,  maybe  omitted  in  a  shorter  course, 
if  thought  desirable.] 


110.  I*Tol>.     To  produce  the  Polar   Equation   of  the  Cissoid  of 
Diodes. 

SOLUTION.— Let  AM,  Fig.  59,  be  the  cissoid,  A,  the  pole,  AX,  the  polar  axis, 
and  P  any  point  in  the  curve  ;  whence  A  P  =  r,  and  P  AX  =  0.     Let  A  B  =  2a. 

Now  AP  =  r=  AD  sec  0  =  D' B  X  sec  0.     But  D'  B  =  E'B*  =  _(gajnn_j)« 

A  B  2a 

=  2a  sin«  0.     Therefore,  r  =  2a  sins  0  sec  0,  or  r  =  2a  sin  6  tan  0.  Q.  E.  D. 

BcfL—Discttssion  of  the  Equation.     For  0  =  0,  r  =  0 For  0  =  45°,  r  = 

a\/2  ;  hence  the  curve  passes  through  C,  Fig.  59 For  6  =  90°,  r  =  oo. 

For  0>90°  and  <270°,  r  is  negative,  and  the  branch,  AM',  is  traced 

while  0  is  passing  from  90°  to  180°,  and  the  branch,  A  M,  is  traced  a  second 

time  by  the  negative  radius  vector  while  0  passes  from  180°  to  270° While 

6  passes  from  270°  to  360°,  r  is  positive  and  AM'  is  traced  a  second  time. 
Therefore  the  curve  is  traced  twice  by  one  revolution  of  the  radius  vector. 


OF   HIGHEK   PLANE   CURVES.  77 

111.  Prob.     To  produce  the  Polar  Equation  of  the  Conchoid  of 

Nicomedes. 

SOLUTION.— In  Fig.  62,  draw  FK  perpendicular  to  OD.  Let  O  be  the  pole, 
and  OD  parallel  to  AX,  the  polar  axis.  Let  AO  =b,  and  AB  =a.  Then  P 
being  any  point  in  the  curve,  we  have  OP=r  =  OF-f-FP  —  OF-f-a.  But 
O  F  =  F  K  X  cosec  FO  K  =  6  cosec  0.  Therefore,  r  =  6  cosec  0  -f  a.  Q.  E.  D. 

Sen. — Discussion  of  the  Equation.      For  0  =  0,  r  =  <x>  ....  For  0  =  90°, 

r  =  b  +  a For  0  =  180°,  r  =  oo For  0  >  180°  and  <  360°,  cosec  0 

is  negative,  and  the  lower  branch  is  traced.  The  student  should  be 
careful  to  observe  the  several  forms,  as  when  a >>  b,  a  =  b,  a<^b.  (See 
85.} 

112.  IP  rob.     To  produce  the  Polar  Equation  of  the  Lemniscate  of 
Bernouilli. 

SOLUTION. — Using  the  same  notation  as  in  Fig.  68,  let  A  be  the  pole,  and  AX 
the  polar  axis.  Let  P  be  any  point  in  the  curve,  and  draw  A  P-  Then  A  P  =  r, 
and  P  AX  =  0.  The  following-  is  an  outline  of  the  solution  : 

F7?2  =  c2  -f-  r2  -f-  2er  cos  0.  (1). 

F~P2  =  c2  -f-  r2  —  2cr  cos  0.  (2). 

Multiplying  (1)  and  (2)  together,  and  remembering  that  F'P  X  FP  =  c2,  we 
have,  after  a  little  reduction  : 

r2  =  4c2  cos2  Q  —  2c2  =  2c2(2  cos2  0  —  1). 
But  2cos2  Q  —  1  =  cos  20.     .  • .  r2  =  2c2  cos  26.    Q.  E.  D. 

Sen. — The  pupil  should  discuss  this  equation  as  the  preceding  have  been. 

[NOTE.— It  is  frequently  more  convenient  to  obtain  the  polar  equation  of  a  curve  by  transforming 
its  rectilinear  equation,  according  to  a  process  to  be  explained  in  a  subsequent  chapter.  We  will 
close  this  chapter  with  some  account  of  Spirals,  a  class  of  curves  of  much  historic  interest  in  con 
sequence  of  the  labor  bestowed  upon  some  of  them  by  the  old  geometricians,  and  to  which  the 
method  of  polar  co-ordinates  is  specially  adapted.] 


OF  PLANE  SPIRALS. 

113.  DBF'S.  —A  Plane  Spiral  is  the  locus  of  a  point  revolv 
ing  about  a  fixed  point,  and  continually  receding  from  it  such  a  man 
ner  that  the  radius  vector  is  a  function  of  the  variable  angle.  Such  a 
curve  may  cut  a  right  line  in  an  infinite  number  of  points,  which 
would  render  its  rectilinear  equation  of  an  infinite  degree.  Hence, 
these  loci  are  transcendental. 

The  Measuring  Circle  is  the  circle  whose  radius  is  the  ra 
dius  vector  at  the  end  of  one  revolution  of  the  generating  point  in  the 
positive  direction. 

A  Spire  is  the  portion  generated  by  any  one  revolution  of  the 
generating  point. 


78 


THE   METHOD   OF   POLAR   CO-ORDINATES. 


1 14.  TJie  Spiral  of  Archimedes  is  the  locus  of  a  point 
revolving  around  and  receding  from  a  fixed  point  so  that  the  ratio  of 
the  radius  vector  to  the  angle  through  which  it  has  moved  from  the 
polar  axis,  is  constant. 


•jo 


11, 


Prob.     To  construct  the  Spiral  of  Archimedes. 

SOLUTION.— Let  A,  Fig.  83,  be  the 
pole,  and  AX  the  prime  radius. 
Through  A  draw  any  convenient  num 
ber  of  indefinite  radial  lines  (say  8) 
making  equal  angles  with  each  other. 
Since  0  and  r  are  to  vary  alike,  when  0 
•=  0,  r  =  0,  and  the  spiral  begins  at  the 
pole.  Take  any  distance,  as  Al,  on 
A  a,  twice  this  distance,  as  A  2,  on  A&, 
three  times  the  same  distance,  as  A3  on 

Ac,  etc.,  etc.     Then  will  1,  2,  3 

17  be  points  in  the  spiral.    Q.  E.  D. 


FIG.  83. 


ILL. — The  dotted  line  abcdefg  is  the  measuring  circle,  and  A123 78  is 

the  first  spire.     8  9  10  11 15  16  is  the  second  spire. 

Sen.— The  several  spires  of  this  spiral  are  parallel,  the  distance  between 
any  two  consecutive  ones  measured  on  the  radius  vector  being  the  same, 
and  equal  at  all  points  to  the  radius  of  the  measuring  circle. 


116.  IProb.     To  produce  the  equation  of  the  Spiral  of  Archimedes. 

SOLUTION. — Letting  a  be  the  ratio  of  the  radius  vector  to  the  variable  angle,  we 
have  r  ='aQ.  Or,  otherwise,  calling  the  radius  of  the  measuring  circle,  A 8, 
Fig.  83,  1,  for  this  value  of  r,  0  =/27f.  Let  3  be  any  point  in  the  curve,  whence  A3 
represents  r,  and  3 AX,  or  8abc  =  0.  Now  from  the  definition,  r  :  1  :  :  0  :  lit. 

Q 
•••r=-.    Q.E.D. 

117.  Con.— The   Reciprocal    or   Hyperbolic   Spiral. 

This  Spiral  is  naturally  suggested  by  the  Spiral  of  Archimedes,  as  in  it  the 
radius  vector  varies  inversely  as  the  variable  angle.     Hence  the  equation  is 


CONSTBUCTTON. — To  construct  the  Recip 
rocal  Spiral,  let  A  be  the  pole,  and  AX 
the  polar  axis.  •  Draw  any  convenient 
number  of  radial  lines  through  the  pole, 
making  equal  angles  with  each  other. 
Take  Al  any  convenient  length,  A 2  = 
£A1,  A3  =  £A1,  A4  =  ^A1,  etc.,  etc. 
The  points  1  2  3  4  -  -  -8  ---  B  are  points 


FIG. 


OF   HIGHER   PLANE    CURVES. 


79 


in  the  curve.  Since  r  can  become  0  only  when  0  =  GO,  this  curve  continues  to  ap 
proach  the  pole  as  the  radius  vector  revolves,  but  reaches  it  only  upon  an  infinite 
number  of  revolutions. 


118.  TJie  Lituus. — The  equation 

of  this  spiral  is  r  =  — j-.     Let   the   stu- 
0* 

dent  construct  it  and  give  the  formal  FIG.  85. 

definition.     The  form  of  the  curve  is  given  in  Fig.  85. 

119.  The  Logarithmic  Spiral. — In  this  spiral  the  radius 
vector  increases  in  a  geometrical  ratio,  while  the  variable  angle  in 
creases  in   an  arithmetical  ratio.     The  equation  is,  therefore,  r=  aO. 
If  a  be  the  base  of  a  system  of  logarithms,  this  equation  becomes  0  = 
logr. 

CONSTRUCTION. — To  construct  r  =  a",  let 
a  =2.  Then  for  6  =  0,  r=l,  which  gives 
the  point  0.  For  0=1,  i.  e.,  the  arc  of 
57.3°  nearly,  r=2l=2,  which  gives  the 
point  1.  For  6  =  2,  t.  e.,  the  arc  of  114.6° 
nearly,  r  =  22  =  4,  which  gives  the  point 
2.  As  0  increases  r  increases  much 
more  rapidly,  so  that  with  this  small  base 
(a  =  2),  at  the  end  of  the  first  revolution, 
when  0=6.28  +  ,  r=  26-28+=more  than  64. 
Hence,  at  one  revolution  the  radius  vector 
would  be  64  times  Ao.  But,  though  r  in 
creases  so  very  rapidly,  it  is  easy  to  see  that 
it  does  not  become  oo  till  6  =  oo .  Again, 
letting  the  radius  vector  revolve  in  the  nega 
tive  direction  from  AX,  so  that  0  is  negative, 
we  have  for  0  =  —  1  r=OAa,  r  =  Aa  =  2-1 
=  £.  For  Q  =  —  2,  r  =  2-2  =  i.  Thus,  it 
appears  that  as  the  radius  vector  revolves  in  this  direction  it  generates  a  portion  of 
the  spiral  which  at  first  rapidly  approaches  the  pole,  but  cannot  reach  it  till 

6  —  oo Were  we  to  take  a  =  10,  the  base  of  the  common  system  of  logarithms, 

the  change  of  r  would  be  so  rapid  that  we  could  represent  but  a  small  arc  of  the 
curve. 


CEAPTEE  EL 

TRANSFORMATION  OF  CO-ORDINATES. 


SECTION  I. 
Methods  of  Passing  from  one  Set  of  Rectilinear  Axes  to  Another, 

120.  DEF'S. — Transformation  of  Co-ordinates  is  the 

process  of  changing  the  reference  of  a  locus  from  one  set  of  axes  to 
another,  or  from  one  system  of  co-ordinates  to  another.  The  prob 
lem  presents  itself  under  two  different  aspects  which  are  nearly  the 
converse  of  each  other  :  1st,  Having  given  the  equation  of  a  locus  re 
ferred  to  one  set  of  axes,  or  system  of  co-ordinates,  to  find  the  equa 
tion  of  the  same  locus  when  referred  to  another  set  or  system.  2nd, 
Having  given  the  equation  of  a  locus  referred  to  some  known 
axes,  to  find  the  position  of  a  new  set,  to  which,  when  the  locus  is  re 
ferred,  its  equation  will  take  some  specified  form.  The  axes,  or  sys 
tem,  to  which  reference  is  made  in  the  given  equation,  may  be  called 
the  Old,  or  Primitive,  Axes  or  System,  and  those  to  which 
the  transformation  is  made,  the  New. 


ILL'S.  — The  equations  x-  -|-  y~  —  25,  y'2 
—  .r(10  — a),  and  x(x  —  4)  -\-y(y  —  6) 
=  12,  may  all  be  considered  as  equations 
of  the  same  locus  M  O  N ,  Fig.  87,  but 
referred,  respectively,  to  the  three  pairs 
of  axes  X,X,  ,  Y,Y,'  ;  X,X,  , 
Y2Y2  ;X,X<  ,  Y3Y3'.  Now,  having 
given  any  one  of  these  equations  any  oth 
er  of  them  may  be  found  if  we  know  the 
position  of  the  new  axis  with  reference  to 
the  eld.  The  process  is  transformation. 

Again,  we  are   familiar  with  various  JYz'        Yf 

methods  of  designating  particular  points  FIG.  87. 

on  the  earth's  surface,  as  by  latitude  and  longitude,  or  by  their  distances  and  direc 
tions  from  a  given  point.  For  example,  we  may  give  the  position  of  Chicago  by 
stating  its  latitude  and  longitude  with  reference  to  the  meridian  of  Washington,  or 


FROM   ONE   RECTILINEAR   SYSTEM   TO   ANOTHER. 


81 


by  giving  its  distances  from  the  tropic  of  Cancer  and  the  meridian  of  Greenwich, 
Eng.,  or,  in  still  another  way,  by  giving  its  distance  and  direction  from  New  York 
city.  The  process  of  converting  one  of  these  descriptions  into  any  other  of  them, 
would  furnish  an  analogy  to  the  process  of  transformation  of  co-ordinates.  The 
first  two  descriptions  (considering  the  earth's  surface  a  plane),  would  be  equations 
of  a  point  (Chicago)  referred  to  rectangular  co-ordinates  ;  the  last  would  be  an  ex 
ample  of  polar  co-ordinates. 

We  will  give  one  more  illustration,  as  it  is  of  the  highest  importance  that  the  na 
ture  of  the  problem  be  understood  from  the  outset  Let  the  student  construct  a 
pair  of  rectangular  axes,  XiXi', 
Y !  Y  i ',  with  the  origin  A  i,  and  an 
other  pair,  X2X2',  Y2  Y2',  also  rect 
angular,  with  the  origin  at  A2  [the 
point  (0,  —1)  when  referred  to  the 
first  axes],  and  the  new  axis  of 
x,  X2X.2',  making  an  angle  with 
the  primitive  axis  of  —  45°,  and  the 
new  axis  of  y  making  an  angle  of 
45°.  Now,  upon  the  first  axes,  con 
struct  x2  —  6xy  -f  f/2  _  QX  _|_  2y  -\-  5 
=  0,  and  upon  the  second  axes  con- 
struct  t/2  —  2ar2  =  2,  when  the  two 
equations  will  be  found  to  give  the  FIG.  88. 

same  locus.  The  problem  of  transformation  which  affords  this  illustration  may 
be  stated  thus  ;  To  transform  x2  —  §xy  -}-  2/2  —  6.r  -f-  2y  -f-  5  =  0,  to  a  new  system 
of  rectangular  axes  having  the  new  origin  at  (0,  — 1),  and  the  new  axis  of  x  mak 
ing  an  angle  with  the  primitive  whose  tangent  is  — 1. 

To  illustrate  the  second  form  under  which  the  problem  of  transformation  pre 
sents  itself,  the  example  of  the  last  paragraph  may  be  stated,— Having  given  the 
equation  x2  —  6xy  +  2/2  —  &e  -f-  2y  -f-  5=.0,  as  the  equation  of  a  locus  referred  to 
rectangular  axes,  required  to  find  the  position  of  a  new  pair  of  axes  to  which,  when 
the  locus  is  referred,  its  equation  will  involve  no  terms  in  the  first  power,  or  in  the 
rectangle  of  the  variables.  The  result  of  the  solution  of  this  problem  would  be  the 
determination  of  the  position  of  new  axes  as  given  in  the  paragraph  above. 


Sen. — As  the/orra  (not  the  degree)  of  the  equation  of  a  locus,  depends  in 
a  large  measure  upon  the  situation  of  the  axes,  or  upon  the  system  used,  it 
will  be  readily  seen  that  a  set  of  axes  in  some  particular  position,  or  some 
particular  system  of  co-ordinates,  may  be  best  suited  to  one  class  of  prob 
lems,  or  of  loci,  and  another  set  or  system  to  another  class.  It  is  therefore  de 
sirable  to  be  able  to  pass  at  will  from  any  one  set  or  system  to  any  other. 
This  transformation  is  effected  by  finding  the  values  of  the  co-ordinates  in 
the  given  equation  in  terms  of  new  co-ordinates  (and  certain  constants)  and. 
substituting  the  latter  for  the  former.  The  methods  of  doing  this  we  will 
now  explain. 


82 


TRANSFORMATION   OF   CO-ORDINATES 


122.  Prob.—  To  produce  the  general  formulae  for  passing  from  one 
set  of  rectilinear  co-ordinates  to  another. 

SOLUTION. — Let  P,  Fig.  89,  be  any 
point  in  a  locus  M  N  referred  to  the 
Primitive  Axes  A ;  X , ,  A  i  Y  l ,  the  co-or 
dinates  being  A  i  D  =  x,  and  P  D  =  y- 
Let  A2X2,  A2Y2  be  the  New  Axes, 
the  co-ordinates  of  the  point  P,  when 
referred  to  them  being  A2D'=#2, 
and  P  D '  =?/2 .  Let  the  angle  included 
between  the  primitive  axes,  Vi  A  ,  X  i , 
be  ft  ;  the  angle  which  the  new  axis  of 
x  makes  with  the  primitive,  X2IXi, 

be  a  ;  the  angle  which  the  new  axis  of  y  makes  with  the  primitive  axis  of  x, 
Yo  rXt  be  a  ;  and  the  co-ordinates  of  the  new  origin,  Aj,  bo  AjG  =m,  A2G 
=  n.  The  problem  now  is,  to  find  the  values  of  the  primitive  co-ordinates  x,  y,  in 
terms  of  the  new  co-ordinates  x2,  y2,  and  the  constants  m,  n,  a,  a',  and  ft,  so  that 
the  latter  may  be  substituted  for  the  former  in  the  equation  of  a  locus  referred  to 
the  primitive  axes,  and  the  equation  be  thus  transformed  and  made  to  represent 
the  same  locus  in  terms  of  the  new  co-ordinates  ;  i.  e.,  referred  to  the  new  axes. 

We  have  #  =  A  i  D  =  AiG  +  AfE-{-  D'F.  But  A  t  Q  =  m  ;  and  fr<3m  the 
triangle  A2D'E,  A2E  :  xt  ::  sinAjD'E  :  sinA2ED',  which  becomes 
A2E  :  x2  : :  sin(/5 —  a)  :  sin/5,  since  A2D'E  =  D'ER  —  D'A2E  =  ft — a, 
and  sin  D'EA2  =  sin  D'ER  =  sin/5  (the  sine  of  an  angle  equals  the  sine  of 


its  sxipplement).     From  this  proportion,    A2E  = 


manner,  from  the  triangle  PD'F,  we  have  D'F  = 
these  equivalents  in  the  value  of  x  as  given  above 

X2  sin  (ft  —  a]  -f-  ?/«  Rin  (fi  — 


x2  sin  (ft  —  a) 

sin  ft 
?/2  sin  (ft 


In  the  same 


sin/f 


.     Substituting 


Again,   y  =  PD  —  A2G  -f-  D'E  -f-  PF.     But  A2G  =  n,  and  from  the  tri 
angles  AjD'E,  and  PD'F,  we  have  as  before   D'E=  —  —g-%  and  PF  = 


Hence 


xz  sin  a  -f-  J/t  fiin  of 


(2). 


123.  COR.  1. —  When  the  New  Axes  are  parallel  to  the  Primitive,  the 

formulce  become 

x  =  m  +  xa,  (1) ;  and  y  =  n  -f-  ya,  (2) ; 

since  in  such  a  case  a  =  0,  and  a'  =  /?  ;  whence  sin  <*  =  0,  sin  «;  = 
sin  /3,  sin  (/5  —  «)  =  sin/?,  and  sin  (/?  —  a')  =  sin  0  =  0. 


FIIOM   ONE   RECTILINEAR   SET   TO   ANOTHER.  83 

124.  COR,  2. — To  pass  from  rectangular  axes  to  oblique,  we  have 

x  =  rn  +  x2  cos  a  +  y2  cos  a',   (1) ; 
and  y  =  n  -f-  x2  sin  a  -f-  ya  sin  of',     (2). 

These  follow  readily  from  the  general  formulae  %  observing  that  in  this 
case  ft  =  90°;  wlience  sin  (/? —  a)  =  cos  or,  sin  (/J —  a'')  =  cos  a', 
and  sin  ft  =  1. 

125  •  COR.  3. — To  pass  from  one  set  cf  rectang  idar  axes  to  another  set 
also  rectangular,  but  not  parallel,  we  have 

x  ===  m  -f  x2  cos  a  —  y3  sin  A',  (1)  ; 
arce?   y  =  ii  -f-  X2  sin  a  -f  ya  cos  a,   (2). 

To  deduce  these  from  the  general  formula  (122),  observe  that  (3  =  90°, 
and  a'—a=  90°,  or  a'  =  90°  +  a ;  'whence  sin  /?  =  1,  sin  (/?  —  a)  = 
cos  «,  sin  (ft  —  a)  =  sin  (90°  —  90°  —  or)  =  sin  (—  of)  =  -  sin  a, 
and  sin  <*'  =  sin  (90d  +  a)  =  cos  <*. 

^^(>.  COR.  4. — To  pass  from  oblique  axes  to  rectangular,  we  have 
x,  sin  ( /?  —  or)  . —  Vo  cos  (  ft  —  (X)     ,   . 

x=m  +  -  -^r^-  (1); 

x,  sin  a  -f  v»  cos  a  /ON 

y  =  =  n+     -smTT  -> 

In  this  case  the  a'  —  a  cf  the  general  formula  becomes  90°,  or 
a>  =  90°  4.  «  ;  w/iercce  sin  (/?  --«')=  sin  (/?  -  -  90°  -  -  a) 
=  sin  {—  [90°  —  (ft—  a)]}  =  —  sin  [90°  —  (ft  —  a)]  =  — 
cos  (ft  —  ^),  and  sin  «'  =  sin  (90°  -J-  «)  ==  cos  «. 


'.  COR.  5. —  When  the  origin  remains  the  same,  and  only  the  direc 
tion  of  the  axis  is  changed,  m  =  0,  and  11  =  0,  and  we  have, 

To  pass  from  one  oblique  set  to  another, 

xn  sin  ( ft  —  of)  +  I/,  sin  ( ft  —  of')          ,.,  x 

X    = : '- ,  \1\) 

sin/J 
j*o  sin  or  -\-  7/2  sii 


To  pass  from  rectangular  to  oblique  axes, 

X   =   Xy  COS  a  -f-  t/a  cos  <*'>  C^-a) 

y  =  xz  sin  of  +  2/2  sin  «' ;  (22) 

To  pass  from  one  rectangular  set  to  another, 

x  =  #3  cos  a:  —  y2  sin  or,  (18) 

y  =  x2  sin  of  +  i/a  cos  of ;  (23) 

To  pass  from  oblique  to  rectangular  axes, 

a-,  sin(/? —  a7)  —  ?/2cos(/?  —  or),  . 

sinytf 
j-o  sin  or  +  1/2  cos  of 


84  TBANSFOKMATION  OF  CO-OBDINATES 

Sen.—  In  (14),  (24)  the  angles  involved  are  those  which  the  new  or  rec 
tangular  axes  make  with  the  primitive  or  oblique  axis  of  x.  It  is  some 
times  convenient  to  have  formulae  for  passing  from  oblique  to  rectangular 
axes,  in  which  the  angles  involved  shall  be  those  which  the  primitive  or 
oblique  axes  make  with  the  new  or  rectangular  axis  of  x.  Such  formulas, 
may  be  deduced  from  the  general  formulas,,  but  are  more  readily  obtained 
from  (12),  (22),  as  follows  : 

Multiplying  (12)  by  sin  a',  x  sin  a!  =  x2  sin  a'  cos  a  -f  y2  cos  a'  sin  «', 
Multiplying  (22)  by  cos  a',  y  cos  a'  =  x.t  cos  a'  sin  a.  -j-  3/2  cos  a'  sin  #', 
Subtracting,  #  sin  a'  —  y  cos  a:'  =  x.2  (sin  a:'  cos  a  —  cos  a'  sin  a)  ; 

Whence  since  sin  a  cos  a.  —  cos  a  sin  a  =  sin  (a'  —  a)  we  have 

.r  sin  a'  —  y  cos  ex.' 
*'  =       sinK-a)      •       (lo) 
In  like  manner  eliminating  a?2,  we  have 

?/  cos  a.  —  x  sin  a 
' 


.Ma-  -a)     • 

[NOTE.—  It  will  afford  the  student  an  excellent  geometrical  exercise  to  produce  each  of  the  above 
sets  of  formula,  directly  from  a  figure.  The  forms  in  corollaries.  1  and  2  are  the  most  important, 
and  should  be  fixed  in  memory.] 

Ex.  1.  Assuming  3a?  -f-  5y  =  15  to  be  the  equation  of  a  right  line 
referred  to  rectangular  axes,  find  the  equation  of  the  same  line  re 
ferred  to  parallel  axes  whose  origin  is  at  (1,  2). 

The  equation  is  3x2  +  5y3  =  2. 

Ex.  2.  Assuming  2#  -f  3y  =  6  to  be  the  equation  of  a  right  line  re 
ferred  to  rectangular  axes,  find  the  equation  of  the  same  line  referred 
to  parallel  axes  whose  origin  is  at  (1,  —  2).  Also  to  new  parallel 
axes  whose  origin  is  at  (  —  3,  —  4).  Also  to  new  parallel  axes  whose 
origin  is  (  —  G,  G).  Construct  the  given  equation  and  verify  the  new 
equations  by  constructing,  on  the  same  figure,  the  new  axes,  and 
observing  the  position  of  the  line  as  referred  to  them.  Notice  where 
the  line  cuts  the  several  axes. 

The  equations  are,  1st,  2xt  f  3?/2  =  10  ;  2nd,  2<r2  -f  3y2  =  24  ;  and 
3rd,  2^2  +  37/a  =  0. 

QUERY.  —  Why  do  not  the  coefficients  of  x  and  y  change  in  the  above  transfor 
mations  ? 

Ex.  3.  Construct  the  locus  x*  —  &r  +  y*  -f-  6y  =  0,  upon  rectangular 
axes.  Then  transform  the  equation  by  passing  to  new  parallel  axes 
whose  origin  is  (4,  —  3),  and,  drawing  these  axes  on  the  same  figure, 
construct  the  new  equation  (xf  -f  y^  =  25)  with  reference  to  these 
axes,  observing  that  the  two  equations  give  the  same  locus. 

Ex.  4.  Given  4?/2  -f  ^-  =  36  as  the  equation  of  a  locus  referred  to 
rectangular  axes,  to  transform  to  new  axes  with  the  same  origin,  the 


FKOM   ONE   KECTILINEAR   SET   TO   ANOTHER.  85 

tangents  of  the  angles  which  the  new  axes  of  x  and  y  make  with  the 
primitive  axis  of  x,  being  3  and  — 3,  respectively.  Verify  by  a  con 
struction. 

SUG'S. — The  formula;  are  x  =  x2  cos  a  -\-  ys  cos  a',  and  y  —  x2  sin  a.  -j-  yt  sin  a ' . 
In  this  case  tan  a  =  3,  whence  sin  a  =v/-rd,  and  cos  a  =\/-^.  Also  tan  a  —  —  3, 
whence  sin  a.'  =  v  -•?$,  and  cos  a!  =  —  v/iV-  Introducing  these  values,  the/orm- 
ul(K  become  x  =  \^^(xt  —  2/,),  and  y  ==  v/Sr(*«  +2/2)- 

jP/ie  transformed  equation  is  5.r2s  -f  6#2ya  +  5y22  =  40. 

Ex.  5.  Given  xy  =  16  as  the  equation  of  a  locus  referred  to  rect 
angular  axes,  to  pass  to  new  rectangular  axes  with  the  same  origin, 
the  new  axis  of  x  making  an  angle  of  45°  with  the  primitive  axis 
of  x.  •  f  Equation,  x<?  —  y22  =  32. 

Ex.  6.  By  the  same  transformation  as  in  Ex.  5,  show  that  y4  -f  %*  + 
6#2?/2  =  2,  becomes  xj  +  y^  =  1.  Construct  the  locus,  and  both  sets 
of  axes,  and  observe  the  position  of  the  locus  with  respect  to  the  two 
sets  of  axes. 

Ex.  7.  Given  the  equation  y  =  ax  -f  6  (the  common  equation  of  the 
straight  line),  to  pass  to  oblique  axes  with  the  same  origin. 

SUG'S. — The  formulce  for  transforming  are  x  = 
X2  cos  a  -j-  yt  cos  a',  and  y  =  xs  sin  a  -}-  y2  sin  a.'. 
Substituting  and  reducing,  we  have 

a  cos  a  —  sin  a  h 


sin  a'—  a  cos  a'  '  sm  a  —  a  cos  a 
which  is  the  equation  of  a  right  line  referred  to 
oblique  axes  making  any  angles  (a,  a')  with  the 
primitive  axis  of  x.  Now,  if  we  desire  simply  the 
form  of  the  equation  of  a  right  line  referred  to  ob 
lique  axes,  we  may  consider  the  new  axis  of  x  as 

coinciding  with  the  primitive,  Fig.  90,  and  let  the  new  axis  of  y  maks  any  angle, 
as  ft,  with  this.     Then  a.  =  0,  and  a'  =  ft  ;  whence  the  equation  becomes 

a ^ 6 

sin  ft  —  a  cos  ft    2       sin  ft  —  a  cos  f? 

Again,  letting  the  angle  NTAj  =  alt  a  =  Sm  ***  ;  whence 


sin/;/  —  acos/3 
sin  a, 


sin  a  -i  sin  a, 


gin  Q  _  smal  ^^^  sin  ft  cos  a±  —  cos  ft  sin  a^         sin(yff  — 

'  °OS' 


:sinBCAl5   or  A^  :  &  :  :  sin  (90°  -j-  ai)  :  sin  (/?  —  a,);   whence  AaC  = 
6  sin  (900  -f-  nr,)  ?>cosat  sin  a, 

*.  +  V, 


calHug  A  j  C  =  l>'.     (See  ,54.) 


8G  TRANSFORMATION   OF   CO-OnDINATES 

Ex.  8.  Given  the  equation  y  =  ax  -f  b,  to  find  the  position  of  a  new 
set  of  axes  parallel  to  the  primitive,  to  which,  when  the  locus  is  re 
ferred,  its  equation  shall  have  no  absolute  term. 

SUG'S.  —  Substituting  for  y,  y.,  -f-n,  and  for  x,  x2  -f-  m>  we  have  y2  —  axz  -f-  am 
-{-b  —  n.  Now,  if  the  new  axes  can  be  so  situated  that  am  -\-  b  —  n  =  0,  or  n  = 
am-}-  6,  the  condition  required  will  be  fulfilled.  But  n  and  m  are  the  co-ordinates 
of  the  new  origin,  and  the  condition  n  =  am  -}-  b(n  and  m  being  co-ordinates) 
designates  a  point  in  the  line  y  =  ax  -\-  6.  Hence  the  new  origin  is  to  be  in  the 
line  y  =  ax  +  b.  (See  33,  Sen.  1.) 

Ex.  9.  Transform  A°y2  -f  B-x*  =  An-B-,  to  parallel  axes  with  the  new 
origin  at(  —  A,  0).  Also  to  parallel  axes  with  the  origin  at  (A,  0). 
Also  to  parallel  axes  with  the  origin  at  (  —  m,  —  n).  (See  So*) 

Ex.  10.   Transform  Ay*  -f-  B~x*  =  A*Bn~,  to  oblique  axes  with  the 

same  origin,  such  that  tan  a  tan  a1  =  --  —  ,  and  obtain  the  result  in 

A~ 

terms  of  the  diameters  lying  on  the  new  axes. 

SUG'S.  —  After  making  the  substitutions  and  collecting  terms,  we  have 
(^•-sinso:'  -{-  B-cos°-a')yt2  -f-  (  J.-sin2a  -j-  B*cos'2<x)x2z-}-l2(A'2siv.asina'  -f-  _B2cos<xcosa') 
xtU*  =  A*B2,  (1),  which  is  the  general  equation  of  an  ellipse  referred  to  oblique 
axes,  the  origin  being  at  the  centre.     If  these  axes  are  so  situated  that  tan  a 

sin  a  sin  a  B- 

tan  a   =  ---  =  --  -    A'2  sin  a.  sin  a   -f-  B-  cos  a.  cos  a  =  0,  and  the  term 
cos  a  cos  a  A* 

in    xtyt     disappears,    and    the    equation    is    (A*  sin  *a    -f   B2  cos2  a'}yf2    4- 

(A2  sin2  a  -f  J52  cos2  a}xs*  =  A*B*.     This  is  the  equation  of  the  ellipse  referred 

to  the  axes  required,  but  it  is  not  in  the  terms  required-  /  it  is  the  equation  of  the 

ellipse,    as     BPC,    referred    to    the    diameters 

A[  Bj,  AjDg,  but  is  in  terms  of  the  semi-axes 

AjB,   A,Q,  and  the  angles  a,  a',  which  the 

new   axes   make  with    the  primitive    axis  of  x. 

Thus,  P  being  any  point  in  the  curve,  A  i  E  rep 

resents  »,,    and  PE  y,.      Now  in  this  equation, 

•when  7/2  =0,  xt  becomes  A!  B2.     Hence  calling 

A,  B,  A,,  we  have  A^-  =  -  A^2 

A*  sin*  a  -}-B*  cos2  a 

A~B* 
or  A*  sin*  a  +  B*  cos2  a.  =  —  .     In  like  manner  for  xt  =  0,  yt  becomes  A  z  D  2. 

Hence  calling  A!  D2  B},  we  have  J5,2  =  --  2  -   or  A2  sin2  a'  4- 

A*  sin*  a'  -f-  jft2  cos2  a' 


B2cos-a'  -  -  -'     Substituting  these  values  of  the  coefficients  of  ys2,  and  xz'2, 


•we  have  -jr^yt"  +  ~^x*~  =  A~s'-     Finally,  dividing  by  A2B2,  and  clearing  of 
fractions,  A^y^  +  B^xf  =  A^B^.     Q.  E.  D. 


FROM  ONE  RECTILINEAR  SET  TO  ANOTHER.          87 

7?2 

SCH. — Diameters  so  situated  as  to  make  tan  a  tan  a'  =  —  —  are  Conju 
gate  Diameters,  as  will  appear  hereafter.  Hence  the  equation  of  the  ellipse  re 
ferred  to  conjugate  diameters,  and  in  terms  of  those  diameters,  is  of  the  same 
form  as  the  equation  of  the  curve  referred  to,  and  in  terms  of,  its  axes. 

Ex.  11.  Transform  A^j*  —  B2x2  =  —  A-B-  to  an  oblique  system  with 
the  same  origin,  such  that  tan  a  tan  a'  =  -^,  and  obtain  the  result 
in  terms  of  the  diameters  lying  on  those  axes. 

SUG'S. — The  student  is  expected  to  recog 
nize  this  as  the  equation  of  the  hyperbola 
referred  to  its  own  axes.  The  transforma 
tion  is  in  all  respects  like  the  above,  except 
that  the  diameter  represented  by  BI  is  ira- 
ar/inary,  i.  e.,  does  not  meet  the  real 
branches  of  the  curve,  hence  we  call  AjD*, 

JJlV/_  1,  or  (A^Djj)2  =  —  -#i9-  Tne 
equation  sought  is  A^y^  —  B^x^'2  =  — 
A^Bf.  FIG.  92. 

SCH. — In  each  of  the  two  preceding  examples,  there  were  given,  the  equa 
tion  of  the  locus,  and  the  position  of  the  new  axes,  from  which  to  find  the 
form  of  the  new  equation.  The  converse  of  this  problem  is  important  ; 
i.  e. ,  Given  the  equation  of  the  locus,  and  some  specified  form  of  its  equa 
tion,  to  find  the  position  of  the  new  axes.  Thus,  for  example, — The  origin 
remaining  the  same,  what  must  be  the  position  of  oblique  axes,  to  which, 
when  the  ellipse  is  referred,  its  equation  will  take  the  same  form  as  the 
common  equation.  To  solve  this  problem,  we  first  transform  A-y*  +  B2x*~  = 
A-B*,  to  oblique  axes,  as  in  Ex.  10,  and  obtain  the  form  (1)  in  the  sugges 
tions.  It  then  remains  to  determine  what  values  a  and  a'  must  have,  i.  e., 
how  the  new  axes  must  be  inclined,  to  make  the  equation  take  the  primi 
tive  form.  Now,  the  required  form  has  no  term  in  x.2i/2,  hence  the  coeffi 
cient  of  xty ,  must  be  0,  that  is,  J.2  sin  a  sin  a  -f  B-  cos  a  cos  a  =  0.  From 

sin  a:  sin  a'                                      B2  ..    ,  ,, 

this, =  tan  a  tan  a  — ;  whence  we  learn  that  the  new  axes 

cos  a  cos  a  A* 

must  be  so  situated  that  the  rectangle  of  the  tangents  of  the  angles  which 

they  make  with  the  primitive  axis  of  x  shall  be  —   — ,  in  other  words  they 

must  be  conjugate  diameters.     Putting  the  resulting  equation  in  the  form 

A2  sin2  a.'  4-  J52  cos2  a'  A*  sin2  a  4-  B*  cos2  a 

^    y  2  _i ~      xzz  =  1,  and  making  xt  =  0, 


and   y2  —  0,    successively,    we    find     that    the    new   semi-diameters    are 
_f  and : .     This  equation  and  these 


A  -  sin*  a  4-  B2  cos"  a''  A2  sin2  a  4-  B-  cos-  a 

values  refer  to  any  pair  of  conjugate  diameters,  as  will  appear  hereafter. 


88  TRANSFORMATION   OF   CO-ORDINATES 


Ex.  12.  Find  the  position  of  oblique  axes 
to  which  when  y2  =  %px  is  referred,  the  equa 
tion  will  still  have  the  same  form. 

SUG'S.  —  Passing  to  oblique  axes  in  general,  the  equa 
tion  becomes 


(n  -}-  X2 sin  #  -j-  ?/2  sin  <x')2  =  2p(ra  -j-  a*2  cos  a:  -f-  ?/2  cos  CK')  ; 

or,  expanding  and  collecting  terms  pIG>  93 

sin?a'y22-[-2sina:sinrt'x2?/2  -(-  sin2aa;82-l-2nsma'  2/2+2nsino:|.r2  -j-?i2 — 2pm =0. 


Now,  in  order  to  meet  the  conditions  of  the  problem, 

1st,  There  must  be  no  absolute  term  ;  hence  ?i2  —  2pm  =  0,  (1)  ; 

2nd,  There  must  be  no  term  in  y2  ;  hence  2n  sin  a  —  2p  cos  a  =  0,  (2)  ; 

3rd,  There  must  be  no  term  in  x2yz  ;  hence  2  sin  a  sin  a'  =  0,  (3)  ; 

4th,  There  must  be  no  term  in  x2~;  hence  sin2  a  =  0,  (4). 

If  these  conditions  can  be  fulfilled,  and  we  can  discover  the  position  of  the  new 
axes  which  fulfills  them,  the  problem  is  solved.  We  observe  that  as  there  are  but 
four  conditions,  involving  four  arbitrary  constants,  a,  «',  ra,  and  n,  these  condi 
tions  can  be  fulfilled.  The  first  condition,  n2  —  2pm  =  0,  or  n2  =  2pm,  requires 
the  new  origin  to  be  on  the  curve,  since  n  and  m  bear  the  relation  of  co-ordinates 

to  the  curve.    The  second  condition,  2n  sin  a'  —  2p  cos  a'=Q,  or  -  =  tan  a'=-. 

cos  a'  n 

requires  that  the  new  axis  of  y  make  an  angle  with  the  axis  of  the  curve,  whose 
tangent  is  p  divided  by  the  ordinate  (n)  of  the  new  origin,  which  is  on  the  curve. 
(This  makes  the  new  axis  of  y  a  tangent  to  the  curve,  as  will  afterwards  appear.) 
The  third  condition,  2sin  a:  sin  a'  =  0,  gives  sin  a  =  0,  since  sin  a'  is  not  0.  This 
requires  the  new  axis  of  x  to  be  parallel  to  the  primitive  axis  of  x  (the  axis  of  the 
curve),  and  makes  it  a  diameter.  The  fourth  condition  sin2  a  =  0,  is  fulfilled  by 
the  last,  and  hence  requires  no  further  attention.  If,  therefore,  the  curve  is  re 
ferred  to  any  diameter,  as  A2X2,  and  a  tangent  A2  Y2  at  its  vertex,  the  equation 
becomes  sin2  a'  y  22  =  (2p  cos  a  —  2n  sin  a)xt  ;  or  since  sin  a  =  0,  and  cos  a.  =  1, 

2/2  2  =    .   .t    Ft,  which  is  the  form   required.      Putting    -7-7^-  =  2p:,  we  have 


Sen.  —  The  equation  y^  =  -  -  xz   leaves  the  problem   indeterminate, 
sin2  a' 

inasmuch  as  a.'  is  a  function  of  yt  •  hence  the  new  origin  may  be  anywhere 
on  the  curve.  In  reality  the  problem  furnished  four  arbitrary  constants 
and  required  but  three  conditions  (the  third  and  fourth  being  but  one)  ; 
hence  we  may  impose  another  ;  that  is,  we  may  put  the  origin  where  ice 
please  on  the  curve. 

Ex.  13.  To  transform  A^y*  —  B*x*  =  —  A*B*  so  that  the  hyperbola 
shall  be  referred  to  its  asymptotes,  i.  e.t  to  the  produced  diagonals 
of  the  rectangle  drawn  on  the  axes  of  the  curve. 


FROM   ONE   RECTILINEAR   SET   TO   ANOTHER. 


89 


gUG's._Let  A,X,  and  A^j  be  the 
primitive  axes,  and  the  asymptotes  A,Xe. 
A,Y2  be  the  new  axes.  As  usual,  let 
X,A,X2  =  —  a,  and  X,A,Y2  =  a  . 
Then,  since  C  B  =  B  D  =  B,  A  l  B  —  A, 
A,  C  =  A ,  D  =  \/A*  4-  £-',  sin  a  =  — 


:,  and  cos  a'  = 


sin  a'  = 


the  — 


FIG  94. 


sign  being  given  to  the  value  of  sin  a  since 

a  is  reckoned  around  the  angular  point  A  x  from  left  to  right,  and  sin  a  is  the  sine  of 

a  negative  arc  less  than  90°.     Putting  these  values  in  the  formulae  for  passing 

from  rectangular  to  oblique  axes,  we  have  x  =  (x.,  -j-yz)-"^ »  and  2/  = 


—  xz} 


7? 


v'A*  -f-  B2 

to  be  transformed,  there  results 


Now,  substituting  these  values  of  x  and  y  in  the  equation 


A*  -f- 


Since  — —  is  constant, 

4 


whence  expanding  and  reducing  x.2yf  — 

we  may  represent  it  by  c,  and  write  the  equation  xzyz  =  c.     In  the  case  of  the 

equilateral  hyperbola  c 


— ,  A  being  the  semi-axis. 
2 


Ex.  14.  To  find  a  system  of  oblique  axes  with  the  origin  at  the 
centre,  to  which  when  the  hyperbola  is  referred,  its  equation  will  take 
the  form  xy  =  c. 

SUG'S. _The  common  equation,  A*if~  —  J52.r2  =  —  A2B2,  becomes,  when  -we 
pass  to  general  oblique  axes  with  the  same  origin 


A*  sin2  a   I  yt'2  -f-  2^42  sin  a  sin  a 
—  JB2  cos2  a   I        —  2B-  cos  a  cos  a' 


+  A*  sin2  a 
—  B2  cos2  a 


The  conditions  imposed  are,   A2  sin2  a'  —  B2  cos2  a'  =  0,    and    A2  sin2  a  — 

B2  cos2  a  —  0  ;   whence   tan  a  =  — — ,   and  tan  a'  =      ,  -.     Now,   in  order  that 

A  A 

these  values  should  indicate  the  positions  of  different  lines,  they  must  be  taken 
with  different  signs.     Thus  the  new  axes  are  found  to  make  angles  with  the  prinii- 

T>  T> 

tive  axis  of  x  whose  tangents  are  — — ,  and  — ,  which  relation  characterizes  asymp 
totes.     The  equation  then  reduces  to 

(2 A2  sin  a  sin  a'  —  2B2  cos  a  cos  a']xty2  =  —  A*B2. 

—  B                               A 
But  the  conditions  above  give  sin  a  =  — - —  cos  a  =  — sin  a  — 


\/A-  -f  B'2 


\/A2  +  B2 


B 


, 
,  and  cos  a:   = 


These  values  substituted  in  the  last  equa- 


VM'* 

tion,  give,  after  reduction,  X2y2  = 
before. 


A-    I    B- 


,  which  is  the  same  form  as  found 


90 


TRANSFORMATION  OF  CO-ORDINATES 


Ex.  15.  Letting  x*  —  6xy  +  yz  —  6x  -f  2?/  -f  5  =  0  repiesent  a  locus 
referred  to  rectangular  axes,  required  the  equation  when  the  refer 
ence  is  to  a  new  set  of  rectangular  axes  with  the  origin  at  (0,  —  1), 
and  the  new  axis  of  x  makes  an  angle  of  — 45°,  or  135°,  with  the 
primitive  axis  of  x. 

Suo's.— The  formula?  for  transformation  become,  in  this  case,  x  =  \/\(xz  -f  ?/2), 
and  y  =  \/l(y2  —  «2)  —  !•  The  transformed  equation  is  y*  —  2x2  =  2  (See 
Fig.  88,  and  the  illustration  accompanying  it.) 


SECTION  IL 

Methods  of  Passing  from  Eectilinear  to  Polar  Co-ordinates, 
and  vice  versa, 

128.  frob. — To  produce  the  formulae  for  passing  from  a  Rect 
angular  to  a  Polar  system  of  co-ordinates. 
SOLUTION. — Let  P  be  any  point  in  a  locus 
M  N  referred  to  the  rectangular  axes 
A  ,  X , ,  A  j  Y , ,  the  co-ordinates  of  P  being 
A ,  D  =  x,  and  P  D  =  y,  when  referred  to 
these  axes.  Let  the  pole  of  the  new,  or  polar 
system,  be  A  2,  whose  co-ordinates  are  m  and  n  ; 
and  let  A2X2,  or  A2X2\  be  the  polar  axis 
making  an  angle  «  with  AX,  or  what  is  the 
same  thing,  with  A2K  parallel  to  A,X,. 
Let  the  polar  co-ordinates  of  P  be  A2  P  =  r, 
and  PA2X2,  or  PA2X2'  =  0.  The  angle 
PA2  K  will  be  0  -f-  a  when  the  polar  axis  lies  above  A2  K,  and  0  —  a,  when  the 
polar  axis  lies  below;  hence,  in  general,  PA2K  e=  0  ±  a.  Now  x  = 
A,D  =  A,B-f  A2H.  But,  from  the  triangle  PA2  H,  A2H  =  r  cos  PA2  K  = 
r  cos  (0  ±  a).  .  • . 

x  =  m  -f-  r  cos  (0  ±  a],  (1). 

In  like  manner  y  =  n  +  rsin(0  db  a),   (2),  as  y  —  PD  =  A,,B  +  PH, 
and  PH  =rsin(0  ±  a). 

If  the  pole  is  at  the  primitive  origin,  m  =  0,  n  =  0,  and 

tf  =  rcos(0  ±  a),  (1,); 
and  ?/  — rsin(0  ±  a),  (2L). 

If  the  polar  axis  is  parallel  to  the  primitive  axis  of  x,  a  =  0,  and  the  formula; 
become 

x  =  m  -f-  rcos  0,  (lf)  ; 

y  =  n  -f-  r  sin  0,  (22)  ;  or,  if  the  pole  is  at  the  prim 
itive  origin,  x  =  rcos 0.   (13)  ; 

y  =  rsin0,  (23). 


FIG.  95. 


FEOM  RECTILINEAR  TO  POLAR  CO-ORDINATES,  AND  YICE  VERSA.      91 

129.  Prob.—To  produce  the  formulae  for  passing  from  a  Polar  to 
a  Rectangular  system  of  co-ordinates. 

SoLimoN.-From     %.   95     we     have      PA2    =    VpH'    4.    A2H",     or 

r  _  ^/(y  —  n)2  -f  (z  —  m)2.     From  the  same  triangle  we  have  also  cos  (0  ±  «)  = 

x~m   _===,  and  sin  (B  ±  a)  =  y~n^=.  which  are  the 

^/(y  _  n)2  4.  (a;  —  m)*  v(j  —  ™'2  +  V'c  ~  m^a 

formulce  sought. 

When  the  polar  axis  is  parallel  to  the  primitive  axis  of  x,  the  formula  are 

-  '»)'.    cos  °  =  -7=2==,'    and   sin  °  = 
>/(y  —  nr  -f  <£  —  "^ 

if  at  the  same  time  the  origin  and  pole  coincide,  the  formula! 

' 


are  r 


Ex.  1.  Transform  x*  +  y*=5ax  to  polar  co-ordinates,  the  pole  being 
at  the  origin,  and  the  polar  axis  coincident  with  the  axis  of  x. 

The  equation  is  r  =  5a  cos  0. 

SUG.—  The/ormwtoe  are  x  =  r  cos  0,  and  y  =  r  sin  0. 

Ex.  2.  Transform  (ar»  +  t/2)'2  =  a'O2  —  2/2)  to  polar  co-ordinates, 
the  pole  being  at  the  origin,  and  the  polar  axis  coincident  with  the 
axis  of  x. 

'     The  equation  is  r2  =  as(cos«  0  —  sin2  0)  =  a2  cos  20. 

Ex.  3.  Transform  r2  =  a2  cos  20  to  (>2  +  y2)2  =  a2O*  —  2/2)-  (See 
last  example.) 

SUG.—  First  put  the  equation  in  the  form  r2  =  «2(cos2  0  —  sin2  0). 

Ex.  4.  Under  the  same  conditions  as  above  transform  r2  cos  20  =  a2 
to  &  —  y*  =  a2.  Also  xy  =  a2,  to  r2  sin  20  =  2a2.  Also  a:2  -f  7/2  = 

(2a  _  #)2,  to  r^  cos  10  =  a^.     Also  reverse  these  processes. 

Ex.  5.  To  deduce  from  A^y*  +  &&  =  ^&>  the  Polar  equation  of 
the  ellipse,  in  terms  of  the  transverse  axis  and  eccentricity,  the  pole 
being  at  the  left  hand  focus  and  the  polar  axis  falling  on  the  axis  of 
the  curve. 

SUG's._The  given  equation  being  put  in  the  form  y2  =  (1  —  e*)(Az  —  #*),  and 
the  formulce  for  transformation  in  the  form  x  =  r  cos  0  —  Ae,  and  y  =  r  sin  0,  and 
the  substitutions  made,  we  have 

r»sin2  0  =  (1  —  e2)(.49  —  r2cos20  -f  2^1ercos0  —  AW). 
Expanding  this  and  reducing  to  a  known  form, 


=         _ 
1  _  ei  cos2  0  1  —  e8  cos  0  ' 


92  TRANSFORMATION   OF  CO-ORDINATES. 


Pieces 0(1  —  e2)  -j_  \/ A*(l  —  e*)a(l  —  eacos-0)  -j-  A'e-co&Q^l  —  eg)8 
1  —  e-  cos-  0 


Ae  cos  0(1  —  eg)  =b  \Ai-Hl  —  e2)2  _  ^e  cos  0(1  —  e2)  d=  4(1  —  e-)  __ 
1  —  e2cos20 1  —  e2  cos'2  0 

'ecc^ — — — ^    ~      .     Now  as  neither  e  nor  cos  0  can  exceed  1,  and  as  each  is 
1  —  e*  cos*  0 

generally  less  than  1,  r  is  positive  only  for  e  cos  6  -f- 1  ;  hence  we  may  reject  the  — 

A(l  +  ecos0>(l  —  e2)       Ail  —  e'2) 

sign  in  this  factor  and  write  r  = —T = ^ 

1  —  e-  cos2  6  1  —  e  cos  0 

If  the  pole  is  taken  at  the  right  hand  focus,  x  =  rcosO  -f  Ae,  and  r  =  — — — —  -. 
(See  107  109.} 

[NOTE. — There  are  expedients  by  which  the  algebraic  reductions  in  this  solution  may  be  simpli 
fied  ;  but  as  our  purpose  is  to  exhibit  simply  the  process  of  transformation,  we  do  not  think  best 
to  avoid  the  work  by  indirect  means.  Were  the  object  merely  to  obtain  the  polar  equation,  the 
process  of  (1O7—1O9)  would  be  much  more  simple  and  elegant.] 

Ex.  6.  Deduce  the  polar  equation  of  the  parabola,  r  = , 

1  —  cub  0 

from  y»  =  2pa?.     (See  108.} 

.r3 
Ex.  7.  Transform  the  equation  of  the  cissoid,  t/9  = ,  to  the 

polar  equation,  r  =  — -,  or  r  =  2a  sin  0  tan  0.    (See  110.) 

COS0 


GENERAL  SCHOLIUM. 

The  student  being  now  familiar  with  equations  as  representatives  of  loci, 
is  prepared  to  use  them  as  instruments  for  the  investigation  of  the  properties 
of  their  loci.  But,  in  carrying  forward  this  work,  the  Calculus  renders 
great  assistance,  and  for  many  purposes  is  indispensable.  Therefore  before 
commencing  the  next  chapter,  the  student  must  become  familiar  with  the 
processes  of  differentiating  the  various  kinds  of  explicit  and  implicit  func 
tions  of  a  single  variable,  with  successive  differentiation,  partial  differentia 
tion,  the  development  of  functions  by  Maclaurin's  and  Taylor's  theorems, 
the  evaluation  of  indeterminate  forms,  and  the  theory  of  Maxima  and 
Minima,  as  treated  in  the  first  two  chapters  of  the  second  part  of  this  vol 
ume.  Having  read  those  chapters,  he  can  turn  back  and  resume  his  study 
of  Geometry  at  this  point.  Students  who  do  not  choose  to  study  the  Cal 
culus,  may  complete  their  course  in  this  subject  by  reading  Sections  XIV. 
and  XY.  of  the  next  chapter  in  this  part. 


CHAPTEE  IV, 

PROPERTIES    OF   PLANE   LOCI   INVESTIGATED   BY 
MEANS  OF  THE  EQUATIONS   OF  THOSE  LOCI. 


SECTION  L 

Tangents  to  Plane  Loci, 

(a)     BY  KECTILINEAR   CO-OKDINATES. 

130.  DEF. — Consecutive  Points  on  a  line  are  points  nearer 
to  each  other  than  any  assignable  distance. 

ILL. — As  we  shall  have  frequent  occasion  to  use  this  conception,  it  is  of  the 
utmost  importance  that  the  definition  be  clearly  comprehended.  For  example, 
when  we  speak  of  P  and  P',  Fig.  96,  as  consecutive  points,  we  do  not  conceive 
them  as  absolutely  in  juxtaposition,  i.  e.,  so  near  each  other  that  there  can  be  no 
other  point  between  them  ;  but  we  mean  simply  that  we  are  to  reason  upon  them  as 
nearer  each  other  than  any  assignable  distance.  In  short,  PP'  is  merely  to  be 
considered  infinitesimal  in  the  sense  of  being  less  than  any  assignable  distance. 
So,  also,  when  we  speak  of  PD  and  P'D'as  consecutive  ordinates,  we  mean  that 
D  D '  is  t o  be  treated  in  the  argument  as  less  than  any  assignable  distance — as  infin 
itesimal. 

131.  DEF. — A    Tangent  (rectilinear)   is  a  right  line  passing 
through  two  consecutive  points  of  a  curve. 

Son.  —For  many  purposes,  it  is  more  convenient  to  speak  of  the  two  con 
secutive  points  through  which  a  tangent  passes,  as  one  point,  and  call  it  the 
point  of  tangency :  this,  of  course,  is  necessarily  the  case  when  the  expression 
is  in  finite  terms,  as  the  distinguishing  of  consecutive  points  requires  infin 
itesimals.  The  student  has  become  familiar,  in  Elementary  Geometry, 
•with  the  conception  of  a  curve  as  a  polygon  of  an  infinite  number  of  infin 
itesimal  sides  :  the  prolongation  of  one  of  these  sides  may  be  considered  a 
tangent. 

132.  Con. — A    Tangent  has  the  same  direction  as  the  curve  at  the 
point  of  tangency. 

133.  Prop. — The  first  differential  coefficient  of  the  ordinate  of  a, 
curve  regarded  as  a  function  of  the  abscissa  (—)  is  the  tangent  of  the 
angle  ivhich  a  tangent  to  the  curve  makes  with  the  axis  of  abscissas. 


PEOPEKTIES   OF  PLANE  LOCI. 


DEM.— Let  MN,  Fig.  96,  be  any  plane 
curve  whose  equation  is  y  =/(&).  Let  P  and 
P'  be  consecutive  points,  and  PD  and  P  D' 
consecutive  ordinates.  Then  is  RS,  drawn 
through  P  and  P',  a  tangent.  Draw  PE 
parallel  to  X'X.  Since  P  and  P'  are  con 
secutive  points,  D  D '  and  P  E  are  contem 
poraneous  infinitesimal  increments  of  a;  and 
?/,  respectively;  L  e.,  DD'  represents  dx, 
and  P'E  represents  dy. 


DDr  X 

FIG.  96. 
Now,   letting  STX  be  represented  by  a,  we  have 


tan  a  =  tan  P'  PE  =  ^=r  =  '-/.     Q.  E.  D. 
r*  t.         ux 

Ex.  1.  What  angle  does  a  tangent  to  the  curve  y  — 
the  point  x  =  2,  make  with  the  axis  of  x  ? 


x*  + 1,  at 


do: 


LUTION. 


-•  =  3x2  —  2x,  which,  for  x  =  2  becomes 
dx 


FIG.  97. 


-^7  =  8,  using  (x',  y')  to  designate  the  particular  point 

in  distinction  from  the  general  point  (a;,  y).  .  •  .  The 
tangent  at  the  point  x  =  2,  makes  an  angle  with  the  axis 
of  x  whose  tangent  is  8,  i.  e.,  an  angle  of  82°  52'  30". 
The  figure  is  that  in  the  margin,  in  which  P  is  the 
point  of  tangency,  and  PTX  is  the  angle  whose  tan 
gent  is  8.  (Tan-'8  =  PTX.) 


Ex.  2.  At  what  point  does  the  curve  in  the 
last  example  run  in  a  direction  making  an  angle 
of  45°  with  the  axis  of  #?     At  what  point  does 
it  make  an  angle  of  135°?     At  what  point  is  it  perpendicular?     At 
what  point  parallel  ? 

SUG. — The  direction  of  the  tangent  being  the  same  as  that  of  the  curve,  we  have 
simply  to  find  where  — ,  or  3x2  —  2x  (which  is  its  general  value  in  this  curve) 

equals  1,  0,  — 1,  or  oo,  as  these  are  the  tangents  of  the  required  angles.  Thus  for 
the  first  we  have  to  find  the  value  of  x  which  satisfies  3ar2  —  2.r  =  1.  This  gives 
x  =  I  and  —  i.  Now,  for  x  =  l,  y  =  1  as  is  found  by  substituting  in  the  equa 
tion  of  the  curve.  This  point  is  P'  in  the  figure.  The  curve  also  runs  in  the 
same  direction  at  (  —  i,  $f),  P"  in  the  figure. 

Answers,  The  curve  is  parallel  at  (0,  1),  and  (§-,  ff ).     It  makes  an 


angle  of  135°  at  x  = 


1  db\/—  2 


— ,  which  being  an  imaginary  point 


signifies  that  the  curve  in  this  plane  does  not  make  an  angle  of 
135°  with  the  axis  of  x.  It  is  perpendicular  at  x  =  +  oo,  and 
—  oo  ;  i.  e.,  as  the  curve  extends  to  the  right  or  left  it  approaches 


TANGENTS — BY   RECTILINEAR   CO-ORDINATES.  95 

perpendicularity  with  the  axis  of  x,  but  becomes  perpendicular 
only  at  an  infinite  distance  from  the  origin. 

134.  Sen. — To  determine  at  what  point  on  a  given  curve  a  tangent  must 

be  drawn  to  make  a  given  angle  with  the  axis  of  x,  —or,  what  is  the  same 

thing,  to  find  a  point  at  which  a  curve  has  a  given  direction,— put  the  value 

dy 

~dx  aS  derived  from  the  equation  of  the  curve,  equal  to  the  tangent  of 

the  given  angle  or  direction,  and  solve  this  equation  in  connection  with  the 
equation  of  the  locus.     To  find  where  the  curve  is  parallel  to  the  axis  of  xt 

put  the  value  of  ~  equal  to  0,  and  solve  as  before.     To  find  where  it  is 

perpendicular,  put  the  value  of  J  equal  to  oo,  and  solve  in  the  same  way. 

dx 

Ex.  3.  At  what  point  on  the  curve  y2  =  2#3,  does  a  tangent  make 
an  angle  with  the  axis  of  x,  whose  tangent  is  3  ?  At  what  point  is 
the  curve  parallel  to  the  axis  of  x?  Where  is  it  perpendicular? 
What  is  the  direction  of  the  curve  at  x  =  8?  Construct  the  figure 
and  observe  the  agreement  of  results  therewith. 

Answers,  At  (2,  4);  at  (0,  0);  nowhere;  tan~l(  ±  6).  The  last 
result  indicates  two  tangents  corresponding  to  x  =  8,  one  drawn 
through  (8,  32),  and  the  other  through  (8,  —32). 

Ex.  4.  At  what  point  in  the  curve  y2  =  2x  +  3#2  must  a  tangent  be 

drawn  to  make  with  the  axis  of  #an  angle  whose  tangent  is  1?  1?  2? 

Answers.  The  first  two  are  impossible.     How  does  this  appear  in 

the  solution,  and  how  from  the  locus?     The  tangent  of  the  angle 

is  2  at  (£,  1),  and  at  (—1,  1). 

Ex.  5.  At  what  point  on  y  =  x*  —  3#s  —  24#  -f-  85  is  the  tangent 
parallel  to  the  axis  of  abscissas? 

Am.,  At  (4,  5),  and  at  (—2,  113). 

Ex.  6.  Find  in  the  curve  y  =  a  +  2  (a?  —  b)*,  the  point  at  which  a 
tangent  is  perpendicular  to  the  axis  of  x.  Result,  At  (b,  a). 

Ex.  7.  Under  what  angle  does  y  =  —?L-  cut  the  axis  of  abscissas? 

_L  ~|~-  QC 

SUG'S.— As  this  curve  cuts  the  axis  of  x  at  (0,  0),  the  question  is,  What  is  its 
direction  at  that  point?  Now  ^  =  -L--*  which,  for  x  =  0,  becomes  1. 
.  • .  The  curve  cuts  the  axis  of  x  at  an  angle  of  45°. 

Ex.  8.  Show  that  the  sinusoid  cuts  the  axis  of  x  alternately  at  45° 
and  135°. 

Ex.  9.  What  angle  does  the  focal  tangent  of  the  common  parabola 
make  with  the  axis  of  x  ? 


96 


PROPERTIES   OF  PLANE  LOCI. 


dy 
135.  COR.  —  If  the  axes  are  oblique,  -~  signifies  the  ratio  of  the  sine 

CiX. 

of  the  angle  which  the  tangent  makes  with  the  axis  of  x,  to  the  sine  of  the 


angle  which  it  makes  with  the  axis  of  y,  i.  e., 


ex 


sm 


a] 


(See  34.} 


130.  JProp.  —  Tne  general  equation  of  a  tangent  to  any  plane  curve  is 

oV 

'  -  '"    ' 


FIG.  98. 


in  which  (x',  y')  is  the  point  of  tangency,  and  x  and  j  are  the  current 
co-ordinates  of  the  tangent. 

DEM.— Let  MN,  Fig.  98,  be  any  plane 
curve  whose  equation  is  y  =f(x),  and  let  P 
be  the  point  of  tangency  whose  co-ordinates 
are  x',  y'.  Now,  the  equation  of  any  line 
passing  through  (x,  y')  is  y  —  y'  =  a(x  —  a;') 
(32).  But,  in  order  that  this  line  should  be 
tangent  to  M  N  at  P,  the  tangent  of  the 
angle  PTD,  which  is  represented  by  a  in 

the  formula,  must  be  (-^-,.      Hence,  substitu- 
dx 

ting,  we  have  y  —  y'—  2L(a  —  x").     Q.  E.  D. 

Ex.  1.  What  is  the  equation  of  a  tangent  to  an  ellipse  referred  to 
its  axes? 

SOLUTION.— The  equation  of  the  locus  is  A*y*  -f-  B2x*  =  A*B*  ;  whence  ^  =  — 
— ^,  which  satisfied  for  the  point  (x',  y')  is  —  -^-,.     Substituting  this  value  of  ~ 

in  the  general  equation  of  a  tangent  (136),  we  have  y  —  y'  =  —  -^~,(x  —  x'  )• 

Reducing,   this  becomes  A*yy'  -f  B*xx  —  A*y' 2  +  JS2*'2.      But  as  (x,  y')  is  a 
point  in  the  locus  A*y'2  -f-  &x">  =  A2B*  ;  hence,  finally,  A-yy'  +  B*xx'  =  A*B*. 

QUERIES.— In   -^  = — X,  what  are  x  and  y  co-ordinates  of?    In  y  —  y'  — 

dx  A2u 


(X  _  x')t  what  are  x  and  y  co-ordinates  of?    What  x'  y'  ?     Of  what  degree  with 


JL 

respect  to  the  variables  is  A*yy'  +  B*xx  =  A*B*  ?  Why  should  it  be  of  this 
degree  (35)  ?  What  are  the  variables?  Notice  that  for  the  same  tangent,  x  and  y 
have  all  values,  but  as'  and  y'  have  fixed  values  :  x',  y'  are  general,  i.  e.,  they  rep 
resent  any  point  in  the  locus,  but  they  do  not  represent  alt  points  at  the  same 

time.     If  in  our  deductions  from  y  —  y'  =  C~(x  —  x'),  we  had  chanced  to  find 

^2^2  _|_  _g-?.T2  could  we  have  substituted  for  it  A*B*t  as  we  did  for  A*y'*  -f  B*x'2  ? 
Why  not? 


TANGENTS — BY   BECTILINEAE   CO-ORDINATES. 


97 


Ex.  2.  Produce  the  equation  of  a  tangent  to  3?/2  +  tf2  =  5,  at  x=  1, 
and  construct,  first,  the  tangent  from  its  equation,  and,  second,  the 
curve  from  its  equation. 

SOLUTION. — In  this  locus  —  = 

.     This  is  the  general  value 

% 

of   the    tangent    of   the    angl 
which  a  tangent  to  this  ellipse 
makes  with  the  axis  of  x.     For 
the  particular  point  x  =  1  (for 
which  y  =  =h  1.155,  nearly),  we 

have  —  =  =F  .29  approximately. 
dx' 


FIG.  99. 


Substituting  these  values  in  the  general  equation  of  a  tangent  (130),  we  have 
y  =p  1.155  =  =F  -29(0?  —  1),  or  y  =  =F  .29.-B  db  1.44.  There  are,  therefore,  two 
tangents  to  this  locus  at  x  =  1  ;  one  whose  equation  is  y  =  —  .29.r  -f-  1.44,  and 
another  whose  equation  is  y  =  .29a;  —  1.44.  RS,  in  the  figure,  represents  the 
former;  and  R'S',  the  latter. 

Another  solution  of  this  example  is  obtained  by  observing  that  the  locus 
3^2  _f-  x2  =  5  is  an  ellipse  whose  semi-axes  are  A  =  v/5,  and  B  =  \/f  .  But  the  equa 
tion  of  a  tangent  to  an  ellipse  is  A*yy'  -f-  JB-'aea;'  =  A*l&  ;  whence,  substituting, 
we  have  5yy'  -f-  §xx'  =  */,  or  3yt/'  -f-  «&'  =  5,  as  the  equation  of  any  tangent  to 
this  ellipse.  For  the  points  (1,  ±  I.f55)  this  becomes  T/  =  =F  .29x  ±  1.44,  as 
before. 


Ex.  3.  Deduce  the  equation  of  a  tangent  to  an  hyperbola. 
of  a  circle.     Also  of  a  parabola. 

The  equation  of  a  tangent  to  an  hyperbola  is 


Also 


"  a  circle  is  yy'  +  xx'  —  R*. 

"  a  parabola  is  yy'  =  p(x+  x'). 


Ex.  4.  What  is  the  equation  of  a  tangent  to  the  parabola  i/2  =  Sx 
at  x  =  4  ?  Construct  the  tangent  from  its  equation,  and  then  con 
struct  the  parabola  as  in  Ex.  2. 

For  (4,  6)  the  equation  is  y  =  f  x  +  3. 

Is  there  another  tangent  for  x  =  4  ? 

Ex.  5.  Produce  the  equation  of  a  tangent  to  3y2  —  2#2  =  10,  at 
x  =  4.  Is  there  more  than  one  tangent  ?  Construct  the  figure  as 
above.  Equation,  y  =  ±  .7127^  =t  .8909. 

Ex.  C.  What  is  the  equation  of  a  tangent  to  if  =  4  —  .r2,  at  x==  3  ? 
Why  is  the  result  imaginary  ? 


98  PROPERTIES  OF  PLANE  LOCI. 

X3 

Ex.  7.  "What  is  the  equation  of  a  tangent  to  if  = ,  at  x  =  2  ? 

TC  OC 

Ans.,  y  =  2x  —  2,  and  y  =  —  Zx  +  2. 

Ex.  8.  Show  that  the  equation  of  the  tangent  to  the  Napierian 
logarithmic  curve  (x  =  log  y)  ia  y  ==  y'(x  —  x'  -{-  1).  Observe  that 
the  ordinate  to  this  curve  at  any  point,  is  the  natural  tangent  of  the 
angle  which  the  tangent  to  the  curve  at  that  point  makes  with  the 
axis  of  abscissas. 

Ex.  9.  What  is  the  equation  of  a  tangent  to  an  hyperbola  referred 

y' 

to  its  asymptotes  (xy  =  m)  ?  Am.,  y  = fx  -f  2y'. 

CC 

Interpretation  of  the  equation  y  =  —  —,x  -f-  2y'.      If  this  represents  the  tangent 

x 

ij' 
to  an  hyperbola  referred  to  rectangular  asymptotes,  —  —,  is  the  tangent  of  the 

X 

angle  which  the  tangent  to  the  curve  makes  with  the  axis  of  x  ;  and  2y'  is  the  dis 
tance  from  the  origin  at  which  the  tangent  cuts  the  axis  of  y,  as  in  all  equations 
of  right  lines  referred  to  rectangular  axes.  But  in  this  case  the  hyperbola  is  equi 
lateral,  since  xy  —  m  is  the  equation  of  an  equilateral  hyperbola  when  the  asymp 
totes  are  rectangular  ;  or,  in  other  words,  no  hyperbola  but  an  equilateral  one  has 
rectangular  asymptotes To  interpret 

y  =  —  y—x  -f-  2y'  for  oblique    axes,  we 

X 

observe  that  2y'  is  AO,  Fig.  100;  and  by  . 
making  y  =.  0  we  find  that  the  intercept 
on  the  axis  of  x,  AT,  is  2#'.     Now  the 

coefficient  of  x,  —  - ,  is  the  ratio  of  the 
x' 

Bine  of  the  angle  which  the  line  (tangent) 

makes  with   the  axis  of  x  to  the   sine 

of  the  angle  which  it  makes  with  the 

axis  of  y,  by  (34).      This  fact  accords  FIG.  100. 

with  the   relations   observable  from   the 

.     AO       sinATO  y'        sin  ATO 

figure.     Thus,  m  the  tnangle  AOT,   —  =  ^7&=?  or    _  =  — -^ 

The  minus  sign  is  explained  by  observing  that  the  line  R  S  lies  across  the  1st 
angle  when  P  is  in  this  angle,  and  to  pass  to  this  position  from  that  in  the  funda 
mental  figure,  Fig.  24,  the  angle  NGX  of  that  figure  becomes  ST"X  of  this,  an 
angle  whose  sine  is  +,  and  equal  to  sin  STA.  But,  by  this  change  of  the 
position  of  the  line,  the  angle  G  H  A  of  Fig.  24,  first  diminishes  to  0  and  then 
re-appears  generated  from  left  to  right  and  henc^  is  a  negative  angle.  Therefore 

4-  sin  ATO             y' 
sin  AOT  is  negative,  and  we  have : —  — ;. 

Ex.  10.  Produce  the  equation  of  a  tangent  to  the  locus  y2  =  2x  -f- 
3.r3,  at  x  =  2. 

Result,  There  are  two  tangents,  viz.  :  y  =  rh  |o?  ±  ^-. 


TANGENTS  —  BY  RECTILINEAR   CO-ORDINATES.  99 

Ex.  11.  At  what  angle   does  the   line  y  =  %x  +  1  cut  the   curve 
7,2  =  4tf?  -4ns.,  10°  14',  and  33°  4'. 


SUG'S.  —  Find  the  point  of  intersection  and  the  tangent  to  the  curve  at  this 
point.     Find  the  angle  included  between  this  tangent  and  y  =  %x  -4-  1  by  (36). 

Ex.  12.  At  what  angle  does  ya  =  lOx  interest  #2  +  y2  =  144  ? 

An*.,  71°  0'  58". 

Ex.  13.  At  what  angle  does  25y2  +  IGa;2  =  1GOO  intersect  16?/«  - 
9^-2  =  __  576  ?  .4ns.    61°  58'  37". 


13V.  Prop.  —  The  general  value  of  the  intercepts*  of  the  axes  by  a 
tangent  are 


and 

in-  which  X  is  the  intercept  on  the  axis  of  x,  and  Y  that  on  the  axis  of  y, 
(x',  y')  being  the  point  of  tangency. 

DEM.  -The  equation  of  a  tangent  being  y  —  y'  =  -j^(x  —  a:'),  if  we  find  where 

this  line  cuts  the  axes  by  making  y  =  0  for  the  intercept  on  the  axis  of  x,  and  finding 
the  value  of  x  ;  and  x  =  0  and  finding  the  value  of  y  for  the  intercept  on  the  axis  of 
y,  we  have  the  results  sought.  Thus  for  y  =  0,  and  x  =  X,  we  have  0  —  y'  = 

%,(-£—  x')>  or  x  =  *•—  tfgjp-  For  x  =  0,  and  y  =  Y,  we  have  Y—  y'  = 
(JL_(Q-x'},orY=y'  —  z'|^,.  Q.  E.  D. 

Sen.—  In  solving  an  example  we"  may  either  apply  these  formula;  or, 
first  get  the  equation  of  the  tangent  and  then  make  x  and  y  successively 
=  0.  This  is  but  an  application  of  (26,  1st). 

Ex.  1.  Erom  A^  +  B*&  =  A^B"-,  show  that  a  tangent  to  an  ellipse 

cuts  the  axes  at  X  =  —  ,  and  Y  =  —  \  ;  i.  e.,  If  from  any  point  in  an 
x  y 

ellipse  a  tangent  and  an  ordinate  be  drawn  to  either  axis,  half  that  axis  is 
a  mean  proportional  between  the  distances  of  the  intersections  from  the 
centre. 

*  This  is  an  abbreviated  form  of  expression  for  "  the  distances  from  the  origin  to  where  the 
curve  cuts  the  axis." 


100 


PROPERTIES   OF  PLANE   LOCI. 


'S— In  the  figure,  AT  —  X,  AD  =  x', 
PD=y,  AC  =  Y,  AB=A.  and  AG  =--  B. 

A'2 

Hence  having  obtained  X  =  — ,  we  have  but 

x 

to  put  it  into  the  form  X  :  A  : :  A  :  x',  to  ob 
serve    the    truth   of   the    proposition.      Also 

Y  =  ?1  gives  Y  :  B  : :  B  :  y. 

A* 
138.  SCH. — Since  X  —  «  we  see  that 


FIG.  101. 


the  intercept  of  the  axis  of  x  does  not  depend  upon  the  conjugate  axis  of 
the  ellipse,  so  that,  if  on  the  same  transverse  axis,  different  ellipses  be  drawn, 
the  intercepts  on  this  axis,  by  tangents  corresponding  to  the  same  abscissa  are 
equal  That  is,  if  x'  and  A  remain  the  same,  AT  is  the  same.  From  this 
we  have  a  ready  method  of  drawing  a  tangent  to  an  ellipse  geometrically. 
Thus,  let  it  be  required  to  draw  a  tangent  to  the  ellipse  Fig.  101,  at  the 
point  P.  Draw  a  circle  (a  variety  of  ellipse)  upon  the  same  transverse 
axis.  Draw  the  ordinate  PD  and  produce  it  to  P'.  Draw  a  tangent  to  the 
circle  at  P'.  This  fixes  the  intercept  AT.  Draw  a  line  through  P  and  T 
and  it  is  the  tangent  sought. 

[NOTE— The  student  should  make  himself  perfectly  familiar  with  this,  and  all  methods  given  for 
drawing  tangents  to  loci  geometrically.] 

Ex.  2.   Show  that  in  the  hyperbola  the  intercepts  on  the  axes  made 

by  a  tangent  are  X  =  — ,  and  Y= -,  and  that  the  proposition  in 

Ex.  1,  is  true  also  of  the  hyperbola. 

139.  SCH.— This  principle  also  affords 
a  method  of  drawing  a  tangent  to  an  hy 
perbola  geometrically.  From  the  given 
point  of  taugoncy  P,  let  fall  the  ordinate 
PD  ;  and  upon  the  transverse  axis  HB, 
and  the  abscissa  AD,  draw  semi-circum 
ferences.  From  their  intersection  let 
fall  LT  a  perpendicular  upon  the  axis 
of  x.  Draw  a  line  through  P  and  T  and 


FIG.  102. 


it  is  tangent  to  the  curve  at  P.     Proof.  Drawing  AL  and  LD,  we  have 

AD  (or  x'}  :  AL  (or  A)  :  :  AL  (or  A)  :  AT.     Whence  AT  ==  --   and  is  the 

x' 
intercept  made  by  a  tangent  at  P. 

Ex.  3.  Prove  that,  if  a  tangent  be  drawn  to  a  parabola  at  any  point, 
the  intercept  on  the  axis  of  x  is  equal  to  the  abscissa  of  the  point  of 
tangency. 


SUBTANGENTS — BY  RECTILINEAR   CO-ORDINATES.  101 

140.  Sen.— The  principle  developed  in  the  solu 
tion  of  this  example  affords  the  most  simple  method 
of  drawing  a  tangent  to  the  parabola,  geometrically. 
Let  it  be  required  to  draw  a  tangent  at  P,  Fig.  103. 
Draw   the    ordhmte    PD,    take    AT    =    AD,    and    — — ^ 
through  P  and  T  draw  a  line.      This  will  be  the     / 
tangent  required. 

Ex.  4.   To  find  where  the  tangent  to  y*x  =  FIQ 

4(2  —  x )    (the  witch  of  Agnesi,  the  radius  of 
the  fixed  circle  being  1),  at  x=  2,  cuts  the  axes. 

Results,  It  cuts  the  axis  of  x  at  x  =  2,  and  the  axis  of  y  at  y  =  GO, 
i.  e.,  is  parallel  to  the  latter. 

[NOTE.— Observe  from  the  last  example  that  a  tangent  may  cut  the  curve.] 


141.  Dm.—  The  Subtangent  is  the  portion  of   the  axis  of 
abscissas  intercepted  between  the  foot  of  the  ordinate  from  the  point 
of  tangency,  and  the  intersection  of  the  tangent  with  this  axis  ;  or  it 
may  be  defined  as  the  projection  of  the  corresponding  portion  of  the 
tangent  upon  the  axis  of  abscissas.     In  each  of  the  three  preceding 
figures  DT  is  the  subtangent  corresponding  to  P  as  the  point  of 
tangency. 

142.  Prop.  —  The  general  value  of  a  subtangent  is 


in  which  (x',  y')  is  the  point  of  tangency. 

DEM.—  In  any  of  the  three  preceding  figures  we  have  from  the  triangle    PI 
DT  =  PD  X  cot  PTD.      But  DT  is  subt,  PD  =  j/',  and,  as  tan  PTD  is 

W-  ,  cot  PT  D  is  ^,      .  •  .  suU  =  //,.     Q.  E.  D. 
dx  dy  dy' 

Ex.  1.  What  is  the  value  of  the  subtangent  of  if  =  3#2  —  12,  at 
#  =  4? 


SuU.  —  y'd—  —  3.     The  pupil  should  construct  the  figure,  if  he  is  not  sure  that 
he  fully  comprehends  the  example  without. 

Ex.  2.  Find  the  value  of  the  subtangent  of  the  common  parabola. 
Of  the  logarithmic  curve.  Results,  2x',  and  m  or  1. 


102  "  PROPERTIES   OF   PLANE   LOCI. 

Ex.  3."  What  is 'the  Value 'of  the  subtangent  of  y2  =  2x*  at  x  =  2  ? 

Ans ,  |. 

Ex.  4.  If  upon  the  same  transverse  axis  different  ellipses  be  drawn, 
prove  that  the  corresponding  subtangents  are  equal. 


SUG'S.  — The  general  value  is  Subt.  ='• 


,  a  result  which 


does  not  depend  upon  B.  This  truth  is 
illustrated  in  Fig.  104,  DT  being  the 
common  subtangent  for  all  the  ellipses, 
corresponding  to  the  same  value  of  a?, 
AD.  This  is  essentially  the  same  truth 
as  was  brought  to  light  in  Ex.  1,  Art. 
137. 


FIG.  104. 


Ex.  5.  Find  the  subtangent  to  the  hyperbola  referred  to  oblique 
asymptotes.  Result,  Subt  =  x'. 

Sen. — In  Fig.  100,  P  being  the  point  of  tangency  (x,  y')t  DT  =  Subt.  = 
x  =  AD.  Now  since  PD  is  parallel  to  AO,  PT  =  OP  ;  i.  e.,  The  inter 
cepts  of  a  tangent  to  a  hyperbola  between  the  point  of  tangency  and  the 
asymptotes  are  equal.  This  affords  a  method  of  drawing  a  tangent  when  the 
asymptotes  are  given.  Thus  let  it  be  required  to  draw  a  tangent  at  P, 
Fig.  100.  Draw  PE  parallel  to  AX,  and  take  AT  =  2PE.  Through 
P  and  T  draw  a  line  and  it  is  the  tangent  required. 


143.  J*rop. — The  general  expression  for  the  length  of  a  tangent, 
i.  e.,  for  the  portion  intercepted  between  the  point  of  tangency  and  the 
axis  of  x,  is 

dx>* 


Tan  — 
in  which  (x7,  y')  is  the  point  of  tangency. 

DEM.— In  any  one  of  Figs.  101,  102,  103,  we  have  from  the  right  angled  triangle 


PDT,  PT*  —  PD2  -f  DT?,  or  PT  =  ** 


PD 


4-  DT  .     Now  PT  is  Tan., 

C-~.      Q.  E.  D. 


Ex.  1.  What  is  the  length  of  the  tangent  to  j/z  =  2x  at  x  =  8? 

Ans.,  4v/L 


ASYMPTOTES  —  BY  RECTILINEAR   CO-ORDINATES. 
Ex.  2.  Show  that 
In  the    ellipse,    Tan  = 


—  pif-  +  Ay* 
^  _  p<f—  \ 


4- 


A\f 


In  the  hyperbola,  Tan  = 

In  the  parabola,  Tan  =  -v/p2+  ^;    jp  'representing    the    semi- 
pararaeter  in  each  case. 


144.  T>EY.—An  Asymptote  is  a  line  toward  which  a  curve 
constantly  approaches,  but  under  such  a  law  that  they  will  never 
meet ;  or,  what  is  the  same  thing,  that  they  will  meet  only  at  an 
infinite  distance  from  the  origin. 

An  asymptote  is  also  conceived  as  a  tangent  to  a  curve  at  an  infinite 
distance  from  the  origin,  which  yet  passes  within  a  finite  distance, 
i.  e.,  cuts  one  or  both  axes  making  finite  intercepts. 

ILL.— It  is  quite  common  for  persons  encounter 
ing  this  idea  for  the  first  time,  to  repudiate  it  as 
an  absurdity  ;  but  the  following  illustrations  will 
familiarize  it.  Let  the  law  of  the  curve  be  such 
that,  if  ordinates  Bb,  Co,  Dd,  Ee,  Ff,  Gg,  etc., 
be  drawn  at  equal  distances  from  each  other,  each 
succeeding  ordinate  shall  be  £  the  preceding.  It  is 
evident  that  the  curve  will  continually  approach 
AX  but  under  such  a  law  that  it  can  never 
absolutely  reach  it.  (Practically  such  a  curve  will 
soon  become  indistinguishable  from  the  line,  that 
is,  will  run  into  it.)  AX  is  an  asymptote  to  this 
curve. 

In  a  similar  manner  two  curves  may  approach  each  other  under  such  a  law  that 
the  distance  between  them  shall  constantly  diminish,  and  yet  the  curves  never 
meet.  Such  curves  are  asymptotes  to  each  other.  Our  present  purpose  embraces 
only  rectilinear  asymptotes. 


B    C    D   E    F    G    H 
FIG.  105. 


145.  frob. —  To  determine  whether  a  plane  curve  has  rectilinear 
asymptotes. 

SOLUTION. First  determine  whether  the  curve  has  infinite  branches.     If  it  has 

not  an  infinite  branch  it  cannot  have  an  asymptote,  since  an  asymptote  is  a  tangent 
at  an  infinite  distance  from  the  origin.     Second,  if  there  is  an  infinite  branch, 

determine  the  values  of  the  intercepts  of  the  axes  by  a  tangent,  X  =  x—  y^-,  and 


104 


PROPERTIES   OF   PLANE   LOCI. 


Y  =  y  —  x-r^,  for  x  or  y  =  oc.     It  will  be  necessary  to  observe  whether  both  of 

the  variables,  or  on]y  one  of  them,  vary  continuously  to  infinity,  and  get  the  value 
or  the  intercepts  in  terms  of  that  one  which  does  vary  continuously.  If  now  one 
or  both  of  the  intercepts  thus  evaluated  is  finite,  the  branch  has  an  asymptote. 
If  both  intercepts  are  infinite,  the  curve  has  no  asymptote,  since  the  tangent  at  oo 
does  not  pass  within  a  finite  distance  of  the  origin. 

Having  ascertained  that  the  branch  which  is  being  examined  has  an  asymptote, 
it  remains  to  determine  its  position.  If  the  intercepts  are  both  finite  and  not  0, 
their  values  fix  the  position  of  the  asymptote.  If  one  intercept  is  finite  and  the 
other  infinite,  the  asymptote  is  parallel  to  that  axis  on  which  the  intercept  is  infi 
nite.  Finally,  if  the  intercepts  are  0,  i.  e.,  if  the  asymptote  passes  through  the 

origin,  its  direction  is  determined  by  evaluating  —  for  a  tangent  at  infinity. 

Ex.  1.  Examine  t/3  =  Qxz  -f-  x3  for  asymptotes. 

SOLUTION. — Since  as  x  increases  from  0  positively,  y  increases  continuously  and 
without  limit,  is  positive  and  has  but  one  real  value,  there  is  an  infinite  branch 
extending  in  the  first  angle.  Now  when  x  is  — ,  we  have  y*  =  (to2  —  x3,  which 
gives  positive  values  to  y  till  x3  =  G.tf.  After  x3  >  6arJ,  that  is  after  x  >  G  and 
negative,  y  becomes  negative  and  a  branch  is  found  extending  in  the  3rd  angle  to 
infinity.  Either  of  these  branches  may  have  an  asymptote,  they  may  both  have 
the  same  line  for  an  asymptote,  or  they  may  have  different  asymptotes.  To  deter 
mine  what  the  facts  are  we  find  the  intercepts  made  by  a  tangent.  They  are, 

-  7/3        4x2  -f-  x3  —  6x2  —  x3         


JL  =x  —  y 


4x  -f-  &' 


which  for 


x  =  -f-   oo  =  —  2  ;    and   Y  =  y  —  x- 


X* 


2.T2 


— k,  which  for  x  =  -j-  oo  =  2.    .  • .  The  branch 


.  106. 


in  the  first  angle  has  an  asymptote  which  cuts  the 
axis  of  y  at  2  above  the  origin,  and  the  axis  of  x  at 
2  on  the  left  of  the  origin.  The  equation  of  this 
line  is  y  =  x  -f  2.  Finally,  as  the  intercepts  have 
the  same  values  for  x  =  —  oc  as  for  x  =  -f  oo,  this 
line  is  an  asymptote  to  both  branches  of  this  locus. 
The  curve  is  sketched  in  Fig.  106,  in  which  M  N 
is  the  asymptote. 

Ex.  2.  Show  that  y3  =  a5  —  x3  has  an  asymptote  which  is  common 
to  its  two  infinite  branches,  passes  through  the  origin,  and  makes  an 
angle  with  the  axis  of  x  of  135°. 

Ex.  3.  Examine  ?/2  =  2x  +  3^2  for  asymptotes. 
Ex.  4.  Why  has  7/2  =  x*  —  x*  no  asymptote  ? 
Ex.  5.  Examine  y*  =  ax3  for  asymptotes. 
Ex.  6.  Examine  the  conic  sections  for  asymptotes. 


ASYMPTOTES — BY   RECTILINEAR   CO-ORDINATES. 


105 


SOLUTION.  —  An  ellipse  or  a  circle  can  not  have  an  asymptote  as  neither  has  au 
infinite  branch.     It  -remains,  therefore,  to  examine  the  parabola  and  hyperbola. 


From  y*  =  2px,  we  have       =?  ;  hence  *  =  a:  -  2/      = 


,  which 


is  _  oo  for  y  =  oo.     Again  Y  =  y  —  x-      =  y  —     —  y  which  =  oo  for  y  =  co. 
.  •  .  The  parabola  has  no  asymptote.     To  examine  the  hyperbola,  we  have  from 


-  BW  =  -  AW,  |  =  f  I  ;  hence  X  =  x  -  yfy  =  x  - 


=  f  ,  which 


=  0  for  x  =  ±  oo.     Also   Y  =  y  —  flfrg  =  y  -    ~-  = ,  which  =  0  for 

y  —  -4-  oo.  (In  this  curve  both  x  and  y  vary  continuously  to  infinity,  hence  the 
intercepts  may  be  evaluated  in  terms  of  either.)  .-.  The  hyperbola  has  two 
asymptotes,  and  they  both  pass  through  the  centre.  To  determine  the  direction 

dii        B-x  Bx 

of  the  asymptotes  we  evaluate  -f  =  —  = i or  x  =  ±  <x.     t  or 

ax        Ay        Avx'2  —  A1 

X  =  -j-  oo, 

AV'x*  — 

learn  that  the  asymptotes  are  the  produced  diagonals  of  the  rectangle  described 
upon  the  axes,  as  has  been  stated  before. 

146.  Sen.  1.— If,  at  successive  points 
along  an  infinite  branch  of  a  curve,  we 
draw  tangents,  these  tangents  will  either 
approach  some  limiting  position,  or  they 
will  not.     In  the  hyperbola,  Fig,  107,  it 
is  evident  that  the   successive  tangents 
PT,   P'T',   P"T"  are  approaching  the 
limiting  position  SA.     But  in  the  par 
abola,    Fig.   108,    it  is  equally  evident, 

from  the  way  in  which  the  tangents  are  drawn,  that 
there  is  no  limiting  position  beyond  which  a  tangent 
may  not  pass.  Since  AT  =  AD,  AT'  —  AD', 
AT"  =  AD",  the  point  of  intersection  with  the 
axis  recedes  indefinitely  as  the  point  of  tangency 
passes  to  the  right.  In  a  similar  manner  observing 
the  method  of  drawing  a  tangent  to  an  hyperbola, 
Fig.  102,  it  will  appear  that  as  P  recedes,  the  inter 
section  L  constantly  (but  more  and  more  slowly) 
approaches  E  but  can  never  pass  it ;  and,  consequently,  that  T  can  never 
pass  A,  however  far  P  may  recede.  From  these  considerations,  an  asymp 
tote  is  seen  to  be  the  limiting  position  toward  which  a  tangent  approaches 
as  the  point  of  tangency  recedes  to  infinity. 

147.  SCH.  2. Having  found  the  intercept  on  the  axis  of  ordinates  and 

the  tangent  of   the  angle  which  the  asymptote   makes  with   the  axis  of 
abscissas,  we  can  readily  write  the  equation  of  the  asymptote  by  substitu- 


FIG.  107. 


FIG.  108. 


106 


PROPERTIES   OF  PLANE  LOCI. 


ting  in  y  —  ax  -f  b.     Thus  the  equations  of  the  asymptotes  to  the  hyperbola 
are  y  =  —x,  and  y  = -x ;  or  Ay  =  Bx,  and  Ay  =  —  Bx. 


Ex.  7.  Show  that  y  =  —  x  +  -  is  the  equation  of  the  asymptote 
«  o 

to  y3  =  ax*  —  x3. 

Ex.  8.  Show  that  x  =  2a,  and  y  =  db  (x  +  a)  are  asymptotes  to 

y-(x  —  2a)  =  #3  —  a3. 

Ex.  9.   Show  that  the  axis  of  abscissas  (y  =  Oar)  is  an  asymptote  to 
y(ay  -j-  ^2)  =  fl2(fl  —  a;). 

Ex.  10.  Show  that  x  =  2a  is  the  asymptote  to  the  cissoid  of  Diocles. 
Ex.  11.  Examine  x  =  log  y  for  asymptotes. 
Ex.  12.  Examine  y  =  tan  x  for  asymptotes, 

SUG'S. — As  this  curve  is  not  continuous  in  the  direction  a;  =  oo,  we  evaluate  the 
intercepts  for  y  —  oo,  for  which  x  =  ^TT,  $rt,  f TT,  etc.,  and  — ^TT,  — f  TT,  — f TT,  etc., 

Y  =  sec2  £  =  1  -f-  2/2.     .• .  X  =  x  —       '    •-,  which  for  y  =  oo,=  cc  =  ^TT,  ITT,  f  TT, 

etc.,  and  — ITT,  — fyr,  — %7t,  etc.  Y  —  y  —  x(1  -|-  ?/2),  which  for  y  =  oo,  =  oo. 
Hence  there  are  an  infinite  number  of  asymptotes  parallel  to  the  axis  of  y.  (See 
23,  Ex.  27,  Sen.,  Fig.  18.) 

a3 
Ex.  13.  Examine  y  = — -  -f-  c  for  asymptotes. 

(x  —  by2 

SUG'S. — In  examining  this  locus  it  is 
necessary  to  evaluate  the  intercepts  both 
for  x  =  oo,  and  y  =  oo,  as  there  are  infinite 
branches  running  in  both  directions.  The 
general  form  of  the  locus  is  given  in  the 
figure.  The  equations  of  the  asymptotes 
are  ar  =  6  (the  line  M  N),  and  y  =  c  (the 
line  M'N'). 


148.  SCH.  3. — In  very  many  cases 
there  are  more  expeditious  methods  than 
the  above  for  finding  asymptotes.  It  is 
frequently  the  case  that  a  simple  inspec 
tion  of  the  equation  of  the  curve  will 
determine  the  fact.  Thus,  in  the  last  FIG.  109. 

example,  it  is  evident  that  as  x  increases 

from  0  to  b,  y  increases,  and  when  x  =  b,  y  becomes  oo.  .•.  This  branch 
of  the  curve  approaches  the  line  MN,  parallel  to  the  axis  of  y,  and  at  a 
distance  b  from  it,  under  the  law  required  for  an  asymptote.  So  again 
when  x  passes  x  =  b,  it  is  evident  that  y  grows  less,  and  the  curve  approaches 
the  axis  of  x.  But,  as,  when  x  =  oo,  y  =  c,  this  branch  extending  to  the 


TANGENTS  TO   1'OLAR  CURVES.  107 

right  can  never  come  near  the  axis  of  x  than  y  =  c.     In  like  manner  when 
x  =  —  oo,  we  see  that  y  =  c.     .  • .  MN '  is  an  asymptote. 

Ex.  14.  Determine  by  inspection  the  asymptotes  to  xy  —  m. 
Ex.  15.  Determine  by  inspection  that  x  =  a,  and  y—b,  are  asymp 
totes  to  xy  —  ay  —  bx  =.  0. 

SUQ. — Observe  that  y  =  ,  and  x  = 


'j     IBUU    •*/ • 

x  —  a  y  —  b 

149.  Sen.  4. — An  elegant  method  of  examining  for  asymptotes  consists 
in  expanding  y  =f(x),  or  x  =/(y),  into  a  series,  by  the  binomial  theorem, 
Maclaurin's  formula,  or  some  other  method,  when  such  development  is 
practicable.  This  will  be  best  illustrated  by  an  example  or  two. 

Ex.  16.  Determine  the  asymptotes  of  the  locus  x3  —  xy2  +  at/2  =  0, 
by  developing  y  =/(#). 


Now,  if  we  take  the  first  two  terms  of  this  development  we  have  y  =  =fc  x  ±  -, 

the  equations  of  two  straight  lines.  Comparing  this  value  of  y  with  the  entire 
value,  which  is  the  ordinate  of  the  curve,  we  see  that  as  x  increases  the  terms  fol 
lowing  —  grow  less  and  less,  and  consequently  that  the  ordinate  of  the  right  line 
and  the  corresponding  ordinate  of  the  curve  become  more  and  more  nearly  equal ; 
that  is,  the  curve  is  constantly  approaching  the  lines  y  =  ±  x  ±  ^.  Now  when 

x  =  oo  this  difference  vanishes,  as  all  the  terms  following  --,  become  0.     .  • .  The 

lines  y  =  ±  x  d=  i«  are  such  that  the  given  curve  approaches  them  constantly  but 
reaches  them  only  at  an  infinite  distance,  and  are  therefore  asymptotes.  There  is 
also  an  asymptote  at  x  =  a,  which  can  be  discovered  by  inspection,  as  under  the 
last  scholium. 

Ex.  17.  Show  by  developing  y  =/(#),  that  y  =  =fc  x  are  asymp- 

372 

totes  to  y*  = 

Ex.  18.  Show  by  developing  y  =  f(x),  that  y  =  ±  (x  -f-  a)  are 
x$  _!_  dxz 

asymptotes  to  y*  = . 

x  —  d 

SUG. — The  value  of  y  developed  becomes  y  —  db  #(1  -| —  H — :  ~\->  etc.). 


(5  )  TANGENTS  TO  POLAE  CUEVES. 

.  It  is  found  most  convenient  to  determine  tangents  to  polar 
curves  by  means  of  the  subtangents. 

.  DEF. — TJie  Siibtangent  to  a  JPolar  Curve  is  the 


108 


PROPERTIES   OF   PLANE   LOCI. 


distance  from  the  pole  to  the  tangent,  measured  on  a 
perpendicular  to  the  radius  vector  to  the  point  of 
tangency.  Thus  in  the  figure  let  M  N  be  a  curve 
referred  to  P  as  its  pole,  S  any  point  in  the  curve, 
and  RT  tangent  at  S-  Then  PT  drawn  through 
the  pole,  perpendicular  to  PS  and  limited  by  the  tan 
gent,  is  the  subtangent. 


FlG'  110' 


152.  Prop. — The  general  value  of  the  subtangent  to 
a  polar  curve  is 

Subt.  = 
in  which  r  is  the  radius  vector  and  0  the  variable  angle. 

DEM.— In  the  last  figure  let  R  be  a  point  on  the  curve  consecutive  with  S  (infi 
nitely  near  it),  so  that  the  tangent  R  T  is  to  be  considered  as  coinciding  with  the 
curve  between  R  and  S.  Draw  PR,  and  with  radius  vector  PS  as  a  radius 
draw  arc  SQ,  and  also  with  radius  Pb  =  1,  draw  ab  an  arc  of  the  measuring 
circle.  Then  RQ  —  dr,  since  RQ  is  an  infinitesimal  increment  of  the  radius 
vector,  contemporaneous  with  RS.  So  also  RPS,  or  ab  =  dO.  As  SQ  is 
infinitesimal  it  may  be  considered  a  right  line,  and  it  is  perpendicular  to  PR. 
Again,  as  R  approaches  S,  the  triangle  RQS  approaches  similarity  with  SPT  ; 
and  as  it  is  the  relation  at  Ihe  limit  that  we  seek,  we  are  to  treat  RQS  as  similar 
to  SPT.  Hence  we  have  PT  :  PS  :  :  QS  :  QR,  or  subt.  :  r  : :  QS  :  dr. 
But  from  the  similar  sectors  QPS  and  «Pb.  we  have  QS  =  rdQ,  and  substitu 
ting,  subt.  :  r  : :  rdO  :  dr.  .  • .  subt  =  — .  Q.  E.  D. 

dr 

Ex.  1.  Find  the  value  of  the  subtangent  to  the  spiral  of  Archim 
edes. 


SOLUTION. — The     equation     is     r  —  —  ; 

27T 

dO  -°-™ 

whence    —  =  2?r. 
dr 


" — The  annexed  figure  furnishes  an  il 
lustration.  PT  is  subtangent  for  point  S 
and  —  the  square  of  the  numerical  value  of 
0  divided  by  the  circumference  of  the  circle 
whose  radius  is  PB.  The  numerical  value 
of  0  in  this  instance  2^,  since  in  passing 
from  0  =  0  to  S  the  radius  vector  makes  1§ 
revolutions.  But  one  revolution  =  2?r. 


FIG.  111. 


ASYMPTOTES  TO  POLAR  CURVES.  109 

Hence  PT  =  — - —  —  —  •  lit,  or  somewhat  more  than  1-i  times  the  circumference 
^n        G-i 

of  the  measuring  circle.  If  B  is  the  point  of  tangency,  0  —  %7t,  and  subt  P~T'  = 
2?r,  or  the  circumference  of  the  measuring  circle.  In.  this  spiral  the  subtangent 
varies  as  the  square  of  the  measuring  arc. 

Ex.  2.  Prove  that  in  the  hyperbolic,  or  reciprocal  spiral  the  sub- 
tangent  is  constant.  What  is  its  value,  and  what  the  significance  of 
its  sign  ?  Construct  the  curve  and  tangents  at  a,  few  points,  and 
observe  the  subtangents. 

r\ 

Ex.  3.  Find  the  subtangent  to  the  logarithmic  spiral  (r  =  a  ),  and 
show  that  the  angle  under  which  the  curve  meets  the  radius  vector 
is  constant. 

SUG.— The  tangent  of  the  angle  made  by  a  tangent  to  the  curve  and  the  radius 
vector  is  equal  to  the  subtangent  divided  by  the  radius  vector.  In  this  locus  th6 
tangent  of  the  required  angle  is  the  modulus  of  the  system  of  logarithms  used  in 
constructing  the  spiral. 

. 7*3 

Ex.  4.   Prove  that -  is  tlie  value  of   the  subtangent  to  the 

a*  sin  XO 

lemniscate  of  Bernouilli. 


.  I?TOb. —  To  test  polar  curves  for  rectilinear  asymptotes. 

SOLUTION. — Any  curve  which  continually  revolves  around  the  pole  can  have  no 
rectilinear  asymptote  ;  for  with  respect  to  any  fixed  right  line,  such  a  curve  will 
alternately  approach  and  recede.  But  if  for  some  finite  value  of  0,  r  becomes 
infinite,  the  curve  ceases  to  revolve  around  the  pole,  and  will  have  an  asymptote 
if  the  tangent  at  r  =  oo  passes  within  a  finite  distance  of  the  pole  ;  i.  e. ,  if  the 
subtangent  is  finite.  Q.  E.  D. 

To  construct  the,  asymptote,  we  observe  that  the  asymptote  and  radius  vector  are 
drawn  from  a  point  infinitely  distant,  to  the  extremities  of  a  finite  subtangent, 
and  hence  are  to  be  considered  parallel.  We  therefore  determine  the  value  of  the 
subtangent  for  r  =  oo,  and  drawing  the  radius  vector  for  that  value  of  0  which 
renders  r  =  oo,  erect  the  subtangent,  and  through  the  extremity  remote  from  the 
pole  draw  a  line  parallel  to  this  radius  vector.  This  line  will  bo  the  asymptote. 

Ex.  1.  Test  the  hyperbola  for  asymptotes  by  the  polar  method. 
SOLUTION. —The  polar  equation,  when  the 

pole  is  at    F,  is  r  — -.      Now   for 

e  cos  0  —  1 

cos  Q=-,r==  oo  ;  hence  if  there  be  an  asymp 
tote,  it  is  parallel  to  FR  so  drawn  that 
cos  R  FX  =  -  =  —  ...  From  the  equa- 

dO  fecosQ  —  I)2 

tion  we  have  —  =  — — ; -7  ;   whence 

dr        Ae(l  —  e-  j  BUI  6  FIG.  112. 


110  PROPERTIES  OF  PLANE  LOCI. 

dO          A^(l  —  e')*  (ecos  0  —  1)*          A(l  —  e«) 

subt    =  r:—  = X  — =  — 1  which,  since  for 

dr         (e  cos  0  —  1)2  A  Ae(i  _  es)  siu  9  e  sin  6    ' 


cos  0  =  -,  sin  0  =  -  v/e*  —  1,  =  —  J.  v/e*  —  1  =  —  B.     There  is  therefore  an  asymp 

tote.     To  construct  it  draw  FR  making  RFX  =  cos-»f-Y  FD  perpendicular 

to  R  F  and  =  B,  and  through  D  draw  ST  parallel  to  R  F.  ST  is  the  asymp 
tote.  Moreover,  since  cos  (—  0)  =  cosQ,  there  is  another  asymptote  similarly 
situated  below  the  polar  axis  FX.  Finally,  as  the  angle  which  a  diagonal  upon 

the  axes  of  an  hyperbola  makes  with  the  axis  of  x  is  cos-1  —  .  the  asymp- 


totes  are  parallel  to  these  diagonals  ;  and  since  F  D  =  B,  A  F  =  \/A*  -f-  B*,  and 
the  asymptotes  coincide  with  the  diagonals. 

Ex.  2.  Show  by  the  polar  method  that  the  parabola  has  no  asymp 
tote. 

SUG.  —  In  this  case  for  0  =  0,  r  =  oc  ;  but  for  this  value  of  9  subt  —  oo.     Hence 
there  is  no  asymptote. 

Ex.  3.   Show  that  the  hyperbolic  spiral  has  an  asymptote  parallel 
to  the  polar  axis  and  at  a  distance  2?r  from  it. 

Ex.  4.  Show  that  the  polar  axis  is  an  asymptote  to  the  lituus. 


SECTION  II. 
Normals  to  Plane  Loci, 

(a)    BY  RECTANGULAR  CO-ORDINATES. 

1 54.  DEP. — A  Normal  to  a  plane  curve  is  a  perpendicular  to  a 
tangent  at  the  point  of  tangency. 

J«Tt> .  !*TOp. —  The  general  equation  of  a  normal  to  a  plane  curve  is 

dx' 

y-y=-^(--^ 

in  which  (x',  y')  is  the  point  in  the  curve  to  which  the  normal  is  drawn, 
and  x  and  y  are  the  general  co-ordinates  of  the  normal. 

DEM. — Letting  (x',  y')  represent  the  point  in  the  ctirve  to  which  the  normal  is 
to  be  drawn,  the  equation  of  a  tangent  through  this  point  is  y  —  y'  =  -~-,(x  — «')• 

Again  the  equation  of  any  line  passing   through  (a;',  y')  is  y  —  y'  —  a(x  —  x"). 
Now,  in  order  that  the  equation  for  this  general  line  should  become  the  equation 


NORMALS — BY  RECTANGULAR   CO-ORDINATES.  Ill 

of  a  perpendicular  to  the  tangent,  a  must  =  —  — .     .*.  y  —  y'  = -(x  —  x') 

is  the  equation  of  a  normal.     Q.  E.  D. 

ISO.  COR. — The  general  expression  for  the  tangent  of  an  angle  which 
a  normal  makes  with  the  axis  of  abscissas  is  —  — ,  (x,  y)  being  the  point 
in  the  curve  to  which  the  normal  is  drawn. 

Ex.  1.  Produce  the  equations  of  the  normals  to  the  conic  sections. 

A2i/(  y' 

Results,  Ellipse,  y  —  y'  =  -j~,(x  —  x1) ;  Circle,  y  =  -fx  ; 

Jj"X  QC 

A*u'  y' 

Hyperbola,  y—y'=——-(x—x'};  Parabola,  y— y'=—- (x— x'}. 

Sen. — Observe  that  these  equations  do  not  reduce  to  as  simple  and  sym 
metrical  forms  as  do  those  of  the  tangents  to  the  conic  sections.  The  form 
of  the  equation  of  the  normal  to  the  circle  shows  that  the  normal  of  this 
locus  always  passes  through  the  centre.  It  is,  of  course,  the  radius. 

Ex.  2.  What  is  the  equation  of  the  normal  to  y3  =  2#2  —  x*  at 
*  =  1? 

Answer  :  At  x=  1,  y=  ±  1 ;  hence  there  are  two  points  indicated. 
The  equation  of  the  normal  at  the  former  is  y  =  %x  —  1,  and 
at  the  latter  y  =  —  2x  -f  1. 

Ex.  3.  What  is  the  equation  of  a  normal  to  y2  =  6x  —  5  at  y  =  5  ? 
What  angle  does  this  normal  make  with  the  axis  of  x  ? 

Ex.  4.  At  what  point  in  the  ellipse  whose  axes  are  12  and  8  must  a 
normal  be  drawn  to  make  an  angle  of  45°  with  the  axis  of  x? 

Ex.  5.  At  what  point  in  the  witch  of  Agnesi  must  a  normal  be 
drawn  to  be  perpendicular  to  the  axis  of  x  ?  To  be  parallel  ?  To 
make  an  angle  of  135°  ? 


'.  DEF. — TJie  Subnormal  is  the  projection  of  the  normal 
upon  the  axis  of  x ;  or  it  is  the  distance  from  the  foot  of  the  ordinate 
let  fall  from  the  point  in  the  curve  to  which  the  normal  is  drawn,  to 
the  intersection  of  the  normal  with  the  axis  of  x. 

138.  Prob. — To  find  the  general 
value  of  the  subnormal. 

SOLUTION. — In  Fig.  113  PE  is  the  normal  and 
D  E  the  subnormal  for  the  point  P.  Now  in 
the  triangle  P  D  E,  P  D  =  y,  and  tan  P  E  D  =  M 


(numerically)  tan  P  EX  =  -.    .-.  DE  =  y/, 
ay  dx 

by  trigonometry.  Fia.  113. 


112 


PROPEKTIES   OF  PLANE   LOCI. 


Ex.  1.  Show  that  the  subnormal  to  the  parabola  is  constant  and 
equal  to  half  the  latus  rectum.  How  can  a  tangent  be  drawn  to  the 
parabola,  geometrically,  iipon  this  principle  ? 

Ex.  2.  Show  that  the  subnormal  to  the  cycloid  is  (2r?/  —  y~)  • 


159.  COR.—  Since  DC    = 
=  v/CG 


FIG.  114. 


PG 


2ry  -  -  y~,  the  normal  passes 
tl  trough  the  foot  of  the  vertical  diam 
eter  of  the  generating  circle  for  the 
point  to  which  the  normal  is  drawn. 
Moreover,  since  S  P  C  is  a  right  angle,  the  tangent  passes  through  the 
other  extremity  of  the  vertical  diameter. 

1(»O.  Son.  1.  —  This  principle  affords  a  ready  method  of  constructing  a 
tangent  to  the  cycloid  geometrically.  Let  P  be  the  given  point  through 
which  a  tangent  is  to  be  drawn.  Put  the  generating  circle  in  position  for 
this  point  (94),  and  draw  the  vertical  diameter  SC.  Through  3  and  P  draw 
a  right  line  and  it  will  be  the  required  tangent.  Also  PC  will  be  the 
normal  to  the  curve  at  the  point  P. 

101.  SCH.  2.  —  To  draw  a  tangent  which  shall  make  any  given  angle  with 
the  axis  of  x,  draw  the  generating  circle  on  the  axis  HI,  construct  the  angle 
LHI  =  the  complement  of  the  required  angle,  and  through  L,  the  point 
where  this  line  intersects  the  circumference  of  the  central  generating  circle, 
draw  a  parallel  to  the  base  of  the  cycloid.  Where  this  parallel  cuts  the 
curve  P  is  the  required  point  of  tangency.  Through  this  point  draw 
SPT  parallel  to  HL,  arid  it  is  the  tangent  required. 


162.  Prob.  —  To  find  the  length  of  the  normal,  i.  e.,  the  portion 
intercepted  between  the  curve  and  the  axis  of  x. 

SOLUTION.—  In  Fig.   113,  from  the  right  angled  triangle,  PDE  we  have  PE  = 


Ex.  1.  Find  the  length  of  the  normal  in  each  of  the  conic  sections. 
What  is  it  in  the  circle  ? 

Ex.  2.  In  the  cycloid  the  radius  of  whose  generatrix  is  2,  what  is 
the  length  of  the  normal  at  y  =  1  ?  An*.,  2. 

103.  COR.  —  The  normal  is  but  a  particular  caxe  of  a  perpendicular 
to  a  tangent. 


NORMALS  TO  POLAR  CURVES. 


113 


DEM. — As  y  —  y'  =  -^(x —  x')  is  the  equation  of  a  tangent  at  (x',  y'},  a  perpen 
dicular  to  this  through  the  point  (x",  y")  is  y  —  y"  =  —  ^(x  —  x"}.  Making 

the  point  through  which  this  perpendicular  to  the  tangent  is  to  pass   the  point  of 
tangency,  the    perpendicular  becomes  a  normal,   and  its  equation   is  ?/—?/'  = 

dx' 
—  -j-,(x  —  a:'),  since  in  this  case  x"—  x ',  and  y"  =  y' .     For  a  perpendicular  from 

the  origin  on  the  tangent  we  have  y"  =  0,  x"  =  0   and  y  = —  x 

dy'  ' 

Ex.  1.  Show  that  the  equation  of  a  perpendicular  from  the  focus 
of  the  common  parabola  upon  the  tangent  is  y  = —  (# ±p). 

Ex.  2.  Show  that  the  perpendicular  distance  from  the  focus  of  an 
hyperbola  to  the  asymptote  is  B. 

104,  COR. — The  perpendicular  from  the  focus  of  a  parabola  upon 
a  tangent  meets  the  tangent  in  a  tangent  to  the  curve  at  the  vertex  (the 
axis  of  y). 

DEM.— The  equation  of  a  tangent  is  yy'  =  p(x  -f  x'\  and  of  the  perpendicular 
from  the  focus  upon  this  tangent  y  —  —  V-(x  —  ip).  We  have  now  but  to  find  the 
intersection  of  these  lines.  Equating  the  values  of  y,  we  have  —  y~(x  —  ip)  = 
-(x  +  aj'),  or  —  y'2(o;  —  |p)  =  p»x  +  p"x,  or  since  y'2  =  2px',  —  2pxx'  -f  p*yf  = 

p-z-f  p*x"  \  whence  (p  +  2x')a?  =  0.  Now  as  x  can  not  be  — ,  p  -f  2x'  can  not 
become  =  0.  Therefore  to  fulfill  the  condition  ( p  -f-  2x')n-  =  0,  x  must  =  0. 
.  • .  The  point  of  intersection  is  at  x  =  0,  or  in  the  axis  of  y.  Q.  E.  D. 


(&)  NORMALS  TO  POLAR  CURVES. 

165.  DEF. — The  Subnormal  to  a  polar  curve 
is  the  distance  from  the  pole  to  the  normal,  measured 
on  a  line  perpendicular  to  the  radius  vector  to  the 
point  in  the  curve  to  which  the  normal  is  drawn. 
Thus  E  P  is  the  subnormal  of  the  curve  M  N  corres 
ponding  to  R,  the  pole  being  at  P. 

160.  IProb. — To  find  the  general  value  of  the  sub 
normal  to  a  polar  curve. 


(Jfi  D  C> 

SOLUTION.      PT  =  r '-•      Tan  PTR  =  _  = 
dr  PT 


rV 

dr 


_ 

rdO' 


But  tan  PER 


tan  PTR 


=-  =  -3-.     . ' .  P  E  or  subnormal 


dr' 


FIG.  115. 


Q.  E.  D. 


114  PROPERTIES   OF   PLANE   LOCI. 

Ex.  1.  Show  that is  the  value  of  the  polar  subnormal 

of  the  lemniscate  of  Bernouilli. 

r 
Ex.  2.  Show  that  the  subnormal  to  the  logarithmic  spiral  is  — ,  m 

being  the  modulus  of  the  system  ;  and,  consequently,  that  in  the 
Napierian  logarithmic  spiral  the  subnormal  always  equals  the  radius 
vector. 

.  COR.  1. — The  length  of  a  normal  to  a  polar  curve  is  ( f-  r2 )  . 

Vcl^         / 

168.  COR.  2. — The  length  of  a  perpendicular  from  the  pole  upon  a 

r2 
tangent  is  p  =  


DEM.  —  In  Fig.  115,  let  PS  be  the  perpendicular  from  the  pole  upon  the  tangent, 
and  consequently  parallel  to  the  normal   R  E.     From  the  right  angled  triangle 

r?d0 
PST,   PS  =  PT  X  cosSPT.      But  PT   (the   subtangent)  =  -^-  ;    and 

C08  BPT  -  c 


Whence  p  =  -  -  1—  _  =  ——r-  -  -T.     Q.  E.  D. 


SECTION  III. 
Direction  of  Curvature, 

(a)  BY  EECTANGULAB    CO-OBDINATES. 

d2y 
160.  J?rop. — At  a  point  where  -^-  is  positive,  a  curve  is  concave 

d2v 
upward,  and  where  -=-^  is  negative  the  curve  is  convex  upward. 

DEM.— 1st.  Let  ao  be  the  angle  which  a  tangent  makes  with  the  axis  of  *.  When 
the  curve  is  concave  upward,  as  in  Fig.  116,  it  is  evident  that  as  a;  increases  (as  from 
being  the  abscissa  of  P  to  being  that  of  P' ),  ca  increases.  In  other  words,  if  x  takes 
the  infinitesimal  increment  dx,  the  contemporaneous  infinitesimal  change  in  <»  is 


DIRECTION   OF   CURVATURE. 


115 


-f  dta.  Hence  when  the  curve  is  con 
cave  upward,  dx  and  da)  have  the  same 
sign  (x  and  GO  are  increasing  functions 
of  each  other). 

In  a  similar  manner  it  is  evident  that 
when  the  curve  is  convex  upward,  03 
decreases  as  x  increases  ;  L  e.,  if  x  takes 
the  increment  dx,  the  cortemporaneous 
change  in  GO  is  —  dco. 


d?y 


a.* 

dx 


d  •  tan  GO 
~dx~ 


— ^       Now,   as  sec-  GO  is  always 
dx 

positive,   — -n  is  positive  when  x  and  GO 

are  increasing  functions  of  each  other 
(when  dx  and  dco  have  like  signs),  and 
negative  when  they  are  decreasing  func 
tions  of  each  other.  .'.  +  ^  in(^- 


Fio.  117. 


D   D'  D" 


Fio.  118. 


cates  that  the  curve  is  concave  upward,  and  —  -^  indicates  that  it  is  convex 

upward.     Q.  E.  D. 

Another  Demonstration. — Let  DD',  and  D'D", 
Fig.  118,  represent  consecutive  equal  infinitesimal 
increments  of  a,  then  P'E  and  P"E' represent  con 
temporaneous  infinitesimal  increments  of  y.  Kepre- 
sent  them  respectively  by  dyl  and  dy9.  The  differ 
ence  between  dy,  and  dy2,  is  by  definition  d2t/. 
But  when  a  curve  is  concave  upward  it  lies  above 
its  tangent.  Hence  dy2  >  dy,  and  dyt  —  dyl  — 
-f-  d-y.  On  the  other  hand,  when  the  curve  is  con 
vex  upward,  as  in  Fig.  119,  it  lies  below  its  tangent 
and  dy.z  <Cd//,.  Whence  dt/2  — dyi  =  —  d*y.  A 
similar  inspection  can  easily  be  made  in  all  cases, 
both  when  the  curve  lies  above  the  axis  of  x,  and 
when  it  lies  below,  and  thus  the  universality  of  the 

principle  be  established.     Finally,  the  sign  of  ^ 

is  the  same  as  that  of  d*y,  since  d#2  being  a  square 
is  always  positive. 

170.  COR.  I.— By  a  course  of  reasoning  en 
tirely  similar,  talcing  y  as  the  independent  vari 
able,  it  may  be  shown  that  4-  ^  indicates  that  a  curve  is  concave  to  the 

right   and ^  that  it  is  convex  to  the  right. 

dy* 


FIG.  119. 


116  PKOPEHTIES   OF  PLANE   LOCI. 

./7J.  COR.  2. — A  curve  is  convex  towards  the  axis  of  abscissas  when 

d2y 

y— —  is  positive,  and  concave  ichcn  it  is  negative. 
"dx* 

DEM.— For  points  above  the  axis  of  cc,  y  is  -f-,  and  if  the  curve  is  convex  towards 
the  axis  (downward)  — ^  is  also  -j-  ;   hence  y~ -  is  -f--     For  points  below  the  axis 

y  is  — ,  and  if  the  curve  is  convex  towards  the  axis  (upward)  — ^  is  —  ;  hence 
y—  is  -f~-     Therefore  ?y—^  is  always  -J-  when  the  curve  is  convex  towards  the  axis 

of  x.     In  a  similar  manner  it  may  be  seen  that  y— -9  is  —  when  the  curve  is  con 
cave  towards  the  axis  of  x. 

Ex.  1.    To  discover   whether   x2  +  y'2  =  rz  is   convex   or   concave 
towards  the  axis  of  x. 

SOLUTION.— From  x2  -4-  y*  =  r2,  we  have  — -  =—  — .     This  locus  is,  therefore, 

dx*  y* 

convex  upward  when  y  is  -f-,  and  concave  when  y  is  — .     Hence  it  is  always  con 
cave  to  the  axis  of  x. 

Ex.  2.    Test  the  following  for  direction  of   curvature  :    y  =  b  + 
c(x  -\-  a)2  ;  and  y  =  a'2V x  —  a. 

Results,   The  first  is  concave   upward  ;    and  the   second   concave 
towards  the  axis  of  x. 

Ex.  3.  Test  the  direction  of  curvature  y  =  b  +  (x  —  a)\ 

Results,   From  x  ^>  a  to  x  =  oo  convex   towards   the    axis  of  x. 

From  x  <^  a  to  x  =  a  —  b^,  concave.     From  x  =  a  —  b^  to  x  = 
—  oo,  convex. 

Ex.  4.  Examine  y  =  sin  x  ;  x  =  log  y  ;  y  =  tan  x. 


(b)   BY  POLAE  CO-OKDINATES. 

.  DEF. — A  Polar  curve  is  said  to  be  concave  or  convex  towards 
its  pole  at  any  point,  according  as  the  curve  at  that  point  does,  or  does 
not,  lie  on  the  same  side  of  its  tangent  as  the  pole. 


1Y3*  !*rop» — A    polar   curve   is   concave   toward   the    pole   when 

-  —  is  povitire,  and  convex  u-hrn  —   i*  negative  ;    r  being  the  radius  vector 
i..>  dp 

and  p  tJm perpendicular Jrom  the,  pole  upon  the  tangent. 


SINGULAR   POINTS. 


11' 


DEM. — By  a  simple  inspection  of 
(a)  Fig.  120,  it  will  be  seen  that  r 
and  p  are  increasing  functions  of 
each  other  when  the  curve  and  pole 
lie  on  the  same  side  of  the  tangent ; 


.  dr 

hence    —    is   -4-- 
dp 


In  like  manner 


from  (6)  it  is  seen  that  r  and  p  are 
decreasing  functions  of  each  other 
when  the  pole  and  curve  lie  on  dif 
ferent  sides  of  the  tangent ;  hence  --  is 


(a) 


FIG.  120. 


Sen. — In  applying  this  polar  test  for  direction  of  curvature,  it  is 
necessary  that  the  equation  be  in  terms  of  p  and  r.     If  given  in  r  and  0, 

0  can  be  eliminated  between  the  equation  of  the  curve,  and  p  — 

dr1  \4 


(168). 

Ex.  1.  Examine  the  lituus  (r  =  —  \  with  reference  to  direction  of 
curvature. 

SUG'S.  —From  r  =  -    -^  =  iazQ-5  =  —.     This  substituted  in  p  =          ^ 


/<r* 

(* 


gives  p  = 


Whence  --  = 


(4a<«  -f-  r4) 


.  • .  This  spiral  is  concave 


dp      2a2(4a»  — 
v     ~r~  ~*-w  i 

towards  the  pole  for  values  of  r  less  than  av/2,  and  convex  for  r  >>  a 

s\ 

Ex.  2.  Show  r=  a    is  always  concave  towards  the  pole. 


SECTION  IV. 

Singular  Points, 

17J.  DEF. — Singular  Points  of  curves  are  points  which 
possess  some  property  not  common  to  others.  Of  such  points  we 
shall  notice  :  1st,  Points  of  maxima  and  minima  ordinates  ;  2nd, 
Points  of  inflexion  ;  3rd,  Multiple  points  ;  4th,  Cusps  ;  5th,  Isolated 
or  Conjugate  points  ;  6th,  Stop  points  ;  7th,  Shooting  points. 


118 


PROPERTIES   OF  PLANE  LOCI. 


MAXIMA  AND  MINIMA  ORDINATES. 

.  DBF.  —  An  ordinate  is  at  a  maximum  when  it  is  greater  than 
the  immediately  preceding  and  the  immediately  succeeding  values  ; 
and  at  a  minimum  when  it  is  less  than  the  immediately  preceding  and 
immediately  succeeding  values. 

177.  I*  rob.  —  To  find  the  position  and  values  of  maximum  and 
minimum  ordinates. 

SOLUTION.  —  As  y  =/(#),  this  problem  is  the  ordinary  one  of  maxima  and  minima 
of  functions  of  a  single  variable,  treated  in  the  Calculus.     Hence  we  find  the  values 

of  x  which  render  --  =  0,  as  critical  values,  i.  e.,  values  to  be  examined,  and  at 
ax 

which  the  property  exists,  if  it  exist  at  all.     To  distinguish  between  maxima  and 

d*y 
minima  values  we  have  the  common  test  ;  namely,  -j-  —  characterizes  a  mini- 

d2y 
mum,  and  —  —  '-  characterizes  a  maximum,  subject  to  the  conditions  discussed  in 

the  Calculus.     The  value  or  values  of  y  corresponding  to  the  value  or  values  of  j 
found  as  above,  will  be  the  required  maxima  or  minima  ordinates. 

A  GEOMETRICAL  SOLUTION.  —  If  PD 
is  a  maximum,  it  is  evident  that  at 
the  left  of  P  the  tangent  makes  an 
acute  angle  with  the  axis  of  x,  i.  e. 

-.-  is  -f-  ,  and  at  the  right  --  is  —  .     .  •  . 
-'-  =  0  is  the  point  of  change  from 

€t'<C 

-{-  to  —  ,  or  the  point  of  maximum 


G>  *     * 


--  =  0  locates   also   minimum   ordinates. 
dx 


ordinate.  In  like  manner  at  the  left  of  a  point  of  minimum  ordinate,  as  P', 
--  is  — ,  and  at  the  right  -(-. 
Finally,  since  at  a  point  of  maximum  ordinate  the  immediately  preceding  and  suc 
ceeding  ordinates  are  less,  the  curve  is  concave  downward,  whence  we  have - 

dx* 

characterizing  such  a  point.     But,  at  a  point  of  minimum  ordinate,  the  immedi 
ately  preceding  and  succeeding  values  of  y  being  greater,  the  curve  is  convex 

downward  and  we  have  -\-  — ~  characterizing  this  point. 


17 S.  Sen. — If  only  the  numerical  values  of  the  ordinates  be  considered, 

cfolf  6/'V 

when  P  lies  below  the  axis, will  characterize  a  minimum,  and  +  -~-  a 

dx*  dxz 

maximum.       But  a  numerical   maximum,  if  — ,  is   properly   considered  a 
minimum  ;  and  a  negative  numerical  minimum,  is  properly  a  maximum. 


SINGULAR   POINTS.  —  POINTS   OF   INFLEXION. 


119 


Ex.  1.  Examine  y  =  x3  —  9.r2  -f  24#  -f  1G  for  maxima  and  minima 
ordinates. 

SOLUTION.     ^  ==  3x*  —  ISx  -f  24  =  0.     .•.  x  ==  4  Mid  2,  ^  =  6*  —  18.     For 

dx  dx1 

d*i/ 
x  =  -i,  —  =  6  ;  hence  x  =  4  corresponds  to  a  minimum,  which  value  is  32.     For 

x  =  2,  —  -  =  —  6  ;  hence  x  =  2  corresponds  to  a  maximum,  which  value  is  36. 
da;2 

[NOTE.—  The  student  should  construct  the  locus,  and  notice  the  points.  Also  substitute  values 
for  z  a  little  greater  than  4  and  a  little  less,  and  the  same  for  the  point  x  =  2,  observing  in  the 
results  the  maxima  and  minima  values  of  y.  ] 

Ex.  2.  Find  the  location  and  value  of  maxima  and  minima  ordi 
nates  in  the  following  curves  :  (1),  y  =  x*  —  5#4  +  5.r3  -f  1  ;  (2),  y  = 

afl  —  &p«  —  24*  +  85  ;  (3),  y  =  5(ar  —  a?)  ;  (4),  i/  =  (2oa?  —  a?')*  ; 
(5),  y  =  jr«  —  8*3  +  22^  —  24  +  12;  (6),  y  =  6+(a?  —  a)";  (7),  y== 
J7  (a  —  a:)2  ;  (8),  in  the  logarithmic  curve  ;  (9),  in  the  curve  of  tan 
gents  ;  (10),  in  the  cycloid  ;  (11),  in  the  parabola  ;  (12),  in  the  lem- 
niscate  of  Bernouilli. 


P" 


POINTS   OF   INFLEXION. 

(a)  BY  RECTANGULAR  CO-OKDINATES. 

179.  DEF. — A  Point  of  Inflexion  is  a  point  where  a  curve 
changes  direction  of  curvature  for  continuously  increasing  values  of 
x  or  y.     Such  a  point  is  also  characterized  by  the  fact  that  the  tan 
gent  at  the  point  cuts  the  curve  in  the  point  of  tangency. 

ILL. — In  passing  from  P'  to  P",  the  curve 
M  N  changes  direction  of  curvature,  being 
convex  downward  at  P',  and  upward  at  P". 
The  point  P  at  which  this  change  occurs  is  a 
point  of  inflexion.  The  student  should  not 
confound  a  point  of  inflexion  with  such  a 
point  as  P  in  M  N  .  It  is  true  that  reckon 
ing  along  the  curve  from  M'  to  N'  the  curve 
changes  direction  of  curvature  with  reference  pIG-  j22. 

to  the  axis  of  x ;  but  not  so  in  reckoning  along 

AX.     From  D'  to  D  the  curve  is  both  concave  and  convex  towards  the  axis,  and 
does  not  change  at  P,  but  is  limited  there. 

180.  J?rob. — To  determine  points  of  inflexion. 

SOLUTION. — If  examined  with  respect  to  the  axis  of  x,  since,  when  the  curve  is 

d*y  d?y 

convex  downward  we  have  -4 -,  and  when  concave  downward  —  -— ,  at  the 


N\ 

V     I 

^^ 

\ 

p'M 

^ 

^"^ 

P 

/ 

A 

^ 

D       0        Dr 


120  PROPERTIES  OF  PLANE  LOCI. 

point  of  inflexion  —~  must  change  sign,  and  hence  must  =  0,  or  oo,     .• .  If  there 

be  a  point  of  inflexion  it  is  where  — ^  =  0  or  oo.     Having  determined  this  point, 

either  construct  the  curve  in  the  neighborhood  of  it,  or,  better,  substitute  in  —  a 
value  of  x  &  httle  greater,  and  one  a  little  less  than  the  critical  value,  and  observe 
whether  —  really  does  change  sign  at  the  point  under  consideration. 

The  precaution  in  the  latter  part  of  this  solu 
tion  is  necessary  ;  for,  though  a  varying  quan 
tity  cannot  change  sign  without  passing 
through  0  or  oo,  it  does  not  necessarily  change 
sign  upon  passing  through  these  values.  Thus, 

let  M  N  be  a  curve  whose  equation  is  y  =/(«).     

Now,  as  x  passes  from  the  value  A  D  to  that 

of  A  D',  y  passes  through  0,  but  does  not  change  -pIG 

Us  sign.    In  like  manner  by  referring  to  Fig.  109, 

it  will  be  seen  that  in  the  curve  there  delineated,  y  passes  through  oo  without 

changing  its  sign. 

Ex.  1.  Examine  y  =  6  +  (x  —  a)3  for  points  of  inflexion. 

SOLUTION.     —z  =  G(x  —  a)  =  0,  gives  x  =  a,  as  a  critical  point,  L  e.,  one  which 

d-y  d  y 

•may  have  the  property  sought.     Now  for  x  >>  a,  ^-  is  -j- ;  and  for  x  <C  a,  -^ 

is  — .  Therefore  there  is  a  point  of  inflexion  at  x  =  a.  For  x  =  a,  y  =  b;  hence 
the  point  of  inflexion  is  (a,  6). 

Ex.  2.  Examine  the  following  for  points  of  inflexion  :  a*y=x* — ex* ; 

y  =  x +  36#2  —  2x* x4 ;  y  =  sinx;  y— tan#;  x  —  logy,  the  witch 

of  Agnesi. 


(6)   BY  POLAK   CO-ORDINATES. 

181.  Prob. — To  test  polar  curves  for  points  of  inflexion. 
SOLUTION.— The  equation  being  put  into  the  formp  =/(r),  we  have  seen  that 
for  —   -(-,  the  curve  is  concave  towards  the  pole,  and  for  —  — ,  it  is  convex. 

Therefore  £  =  0,  or  oo,  indicates  a  critical  point.     If  upon  examination    -^  is 
found  to  change  sign  at  this  point,  the  point  is  one  of  inflexion.     Q.  E.  D. 

Ex.  1.  Examine  the  lituus  (r  =  —  J  for  points  of  inflection. 
SOLUTION. — First,  to  put  the  equation  in  the  form  p  =f  (r},  we  have  p  = 


SINGULAR    POINTS  —  MULTIPLE   POINTS.  121 

.     Differentiating  the  equation  of  the  locus,  we  find  —    —   —  —  —  » 

"' 


=  i     =  5V   Substituting,  we  have  p  =  _ 

J.,j  r4  _  4#4 

Differentiating,  -y-  =  —  2a2  --  5  .     Putting  this  equal  to  0,  we  obtain  r4  — 
(r«  +  2a<)k" 

4a4  —  0,  whence  r  =  av'iL      Now,  for  r  >-  a-J~l,^-  is  —  ;    but  for  r  <<  #^2,  -|- 

i  s  -f-  .     There  is  therefore  a  point  of  inflection  at  r  =  a  *S2,  for  which  value  0  = 
28°  38'+. 

ciO* 
Ex.  2.  Examine  r  =  --  -  -  for  points  of  inflexion. 

dr2      4r(r  —  a>3  W2  ,  dp 

SOLUTION.     -—  -  —  --  .     .  •.  p  =  ----  --  -,  ana  —  — 

* 


there  is  a  point  of  inflexion,  it  must  be  where  r  passes  through  0,  fa,  or  fa.     But 

—  changes  sign  only  with  the  factor  6r2  —  13ar  -f-  Ga2  ;  and  this  factor  does  not 
di' 

change  sign  when  r  passes  through  0,  but  does  at  r  =  |a  and  fa.  (To  determine 
these  facts,  substitute  r  =  0  -f  h,  and  r  =  0  —  h  ;  also  r  =  |a  -j-  /t,  and  r  =  fa  —  A, 
etc.,  h  being  treated  as  infinitesimal.)  .  •  .  There  is  a  point  of  inflexion  at  r  —  fa. 
And  also  at  r  =  fa.  Where  r  =  f  a,  B  =  +/3,  or  about  99°.  26. 


MULTIPLE   POINTS. 

182.  DEF. — There    are    two    species  of 
Multiple  Points,  viz.,  1st,  A  point  where 
two  or  more  branches  of  a  curve  intersect ; 
2nd,  A  point  where  two  or  more  branches  are 
tangent  to  each  other.     The  latter  are  some 
times  called   Points  of  Osculation.     The  an 
nexed  figures   illustrate   both  species.     The 
first  curve  has  a  triple  point  of  the  first  species 
at  P,  and  the  second  a  double  point  of  the 
second  species  at  P. 

183.  jPTob. — To   examine   a   curve  for 

m  ultiple  points.  Fia  125. 

SOLUTION.— Since  two  or  more  branches  pass  through  a  multiple  point,  for  x  = 
the  abscissa  of  such  a  point,  y  has  but  one  value,  while  at  other  points  near  it,  y 
has  two  or  more  values  for  each  value  of  x.  In  explicit  functions,  or  in  functions 


122  PROPERTIES  OF  PLANE  LOCI. 

of  a  comparatively  simple  form,  such  a  point  can  generally  be  determined  by 
inspection.  Having  found  a  value  of  x  for  which  y  has  but  one  value,  and  on  both 

dy 

dx 

unequal  values  at  this  point.  If  --  has  unequal  values  the  branches  of  the  curve 
intersect  at  the  point,  since  their  tangents  do,  and  the  point  is  of  the  first  species. 
If  —  has  but  one  value  for  these  values  of  x  and  y,  the  tangents  to  the  branches  at 

the  point  coincide  and  the  point  is  of  the  second  species. 

When  the  critical  points  are  not  readily  determined  by  inspection,  put  the  equa 
tion  in  the  form  of  an  implicit  function  without  radicals.     Let  it  be  u  =f(x,  y)  =  0. 
du 

Form  -r^  =  —  — .     Now.  as  the  equation  of  the  locus  did  not  contain  radicals. 
dx  du 

and  as  differentiation  does  not  introduce  them,  the  only  way  in  which  --  can  have 

du. 

several  values  is  by  taking  the  form  -.     Hence  we  have  -  -  =  —  —  =  -,  or  —  =  0, 

0  dx          du      0        dx 

dy 

and  --  =  0,  from  which  to  determine  critical  values  of  a;  and  y,  (that  is,  those 
dy 

values  which  may  correspond  to  multiple  points).     Solving  the  equations  --  =  0, 

and  ---  =  0,  for  x  and  y,  see  which  of  the  values  found  satisfy  the  equation  of  the 

locus.  If,  at  any  point  thus  determined,  y  has  but  one  real  value  for  the  particular 
value  of  x,  and  on  both  sides  of  it  y  has  two  or  more  real  values,  this  point  is  a 

multiple  point.  Its  species  can  be  determined,  as  before,  by  evaluating  --  =  -,  for 
the  particular  values  of  x  and  y  which  locate  the  point. 

^  Ex.  1.  Test  for  multiple  points  y  =  (x  —  a)  Vx  -f  b. 

SOLUTION.  —Since  \/x  is  both  -f-  and  — ,  y  has  in  general  two  values.  But  it  is 
evident  that  for  x  =  0,  y  has  but  one  value,  namely, 
b  ;  also  for  x  =  a,  y  has  but  one  value,  b.  These 
are  the  critical  values  of  x  and  y.  Upon  the  point 
(0,  6},  we  observe  that  the  branches  do  not  pass 
through  it ;  since  for  x  negative  y  is  imaginary. 
Hence  (0,  6)  is  not  a  multiple  point.  But  upon 
the  point  (a,  6)  we  observe  that  y  has  two  real 
values  on  each  side  of  it.  This  is  therefore  a  / 

double  point.     Now  ~  =  ±  3x  ~a,  which  for  FlG-  126' 


a  gives  —  =  rfc  v/a.     .• .  The  point  is  of  the  first  species,  and  the  tangents 


SINGULAR   POINTS.  123 

to  the  curve  at  the  point  make  angles  with  the  axis  of  x  whose  tangents  are 
+  v'a,  and  —  \/a.  The  form  of  the  curve  is  given  in  the  figure. 

Ex.  2.  Examine  t/2  =  xz  —  x4  for  multiple  points. 

Ex.  3.  Examine  x4  +  2a#2y  —  ay3  =  0  for  multiple  points. 

SOLUTION  .—As  it  is  not  easy  to  discover  by  inspection  all  the  points  to  be  examined 

du 

in  this  case,  we  will  proceed  by  the  second  method.     We  find  --  =  —  —  =  — 

dy 

+  *axy  ^    ^y^ence  4#3_|_  ±axy  =  0,  and  2cuc2  —  3<xt/2  =  0.     These  equations  give 
2a#2  —  3ay2' 

(  x  =  0        <  x  =  ia\/6           ,   j  x  —  —  iav/6~ 
the  following  critical  values :      ~\      _  ;      i      2    ;    and   1      , 

But  of  these  only  the  first  set  satisfy  the  equation  of  the  curve.  The  point  (0,  0) 
is,  therefore,  to  be  examined.  Since  none  but  even  powers  of  a-  are  involved,  a 
change  in  its  sign  does  not  change  the  form  of  the  function  ;  hence  the  form  of 
curve  is  the  same  on  both  sides  of  the  axis  of  y.  As  the  equation  is  a  cubic,  there 
it  at  least  one  real  root,  and  hence  one  branch  at  least  passes  through  the  origin 
in  the  plane  of  the  axes.  To  determine  whether  the  other  roots  are  real  or  imagi 
nary,  and  hence  whether  the  other  branches  lie  in  the  same  plane  with  the  axes 
we  might  solve  the  equation.  But  this  is  not  necessary.  We  can  more  readily 

$u  4T*^  -1-]— 

determine  the  facts  by  examining  the  tangents.     Evaluating  --=  —  -' 


Actx*  —  d(iy~ 
for  x  __  o,  y  =  0,  we  find  --  =  0,  -f-  \/2  and  —  \/2.     Therefore  there  are  three 

tangents,  and  the  point  is  a  triple  point  of  the  first  species.  The  curve  is  that 
given  in  Fig.  124,  (182). 

Ex.  4.  Examine  at/3  —  x3y  —  ax3  =  0  for  multiple  points. 

gUG's.  —The  values  arising  from  —  =  0,  and  —  =  0,  are  x  —  0,  y  =  0,  and  x  = 

dx  dy 

a  ^3,  y  =  —  a.  But  only  the  first  satisfy  the  equation  of  the  curve.  Evaluating 
dy  3x^  +  30.7^0  forthese  yal  we  find  /ugi  for  S\ps=  1,  ps  _i  =0, 

^j.  3ay2_a;3  Q  V  OX/ 

or  (p  _  l)(p2  _j_  p  -|_  1)  =  0.  Whence  p  =  1,  or  —  \-  ±  iv/—  3.  Hence  we  see 
that  there  is  but  one  tangent  in  this  plane,  and  therefore  but  one  branch  passing 
through  the  origin,  and  no  multiple  point. 


Ex.  5.  Show  x4  +  x*y*  —  §ax*y  +  «5y«  =  0  has  a  multiple  point  of 
the  second  species  at  the  origin, 


124 


PROPERTIES   OF   PLANE   LOCI. 


CUSPS. 

184.  DEF. — A  Cusp  is  a  variety  of  the 
second  species  of  double  point,  in  which 
the  osculating  branches  terminate  in  the 
point.  Cusps  are  of  two  kinds  :  1st,  When 
the  branches  lie  on  different  sides  of  the 
tangent ;  2nd,  When  the  branches  lie  on 
the  same  side  of  the  tangent. 


FIG.  127. 


18  5  •  I*Tob. — To  examine  a  curve  ft 


or  cusps. 


SOLUTION.— The  process  is  the  same  as  for  multiple  points  of  the  second  species, 
the  only  difference  being  that  the  branches  stop  at  the  point  instead  of  running 
through  it ;  and  hence  that  the  values  of  y  are  real  011  one  side  and  imaginary  on 
the  other. 

To  ascertain  of  which  kind  the  cusp  is,  we  may  compare  the  ordinates  of  the 
curve  in  the  vicinity  of  the  point,  with  the  corresponding  ordinate  of  the  tangent ; 

or,  by  means  of  ~,  ascertain  the  direction  of  curvature  ;   or  we  may  construct 

the  curve  about  the  point.  By  the  first  method  we  discover  that  the  cusp  is  of  the 
first  kind,  if  the  ordinate  of  the  tangent  is  intermediate  in  value  between  the  cor 
responding  ordinates  of  the  curve  ;  and  that  it  is  of  the  second  kind  if  the  ordi 
nate  of  the  tangent  is  less  or  greater  than  both  the  corresponding  ordiuates  of  the 
curve. 

If  the  common  tangent  is  perpendicular  to  the  axis  of  x,  it  is  best  to  discuss 
tho  cusp  with  respect  to  the  axis  of  y,  using  --.  etc. 

Ex.  1.  Examine  (?/  —  /;)-'  =  (x  —  a)3  for  cusps. 

SOLUTION.— We  have  y  =  b  ±  (x  —  a/,  from 
which  we  see  by  inspection  that  for  x=a,  y  has 
but  one  value,  for  x<^a,  y  is  imaginary,  and  for 
x  >  a,  y  has  two  real  values.  Therefore  (a,  b) 

(Tti                          - 
i.-.  tae  point  to  be  examined.     -  -  =  ^  \  x a) 2     

which  for  x  =  a  becomes  ±  0.     Hence  the  two 

tangents   are   seen   to   coincide,    their   common 

equation  being  y  =  b  ;   and  there  is  a  cusp.     To  determine  the  kind  of  cusp,  we 

consider  the  values  of  the  ordinates  of  the  curve  for  x  a  little  greater  than  a,  as 

a  +  h,  h  being  an  infinitesimal.     Substituting,  we  have  y  =  bd=(a--(-h af  — 

b±:h'2.    Thus  we  see  that  one  of  the  ordinates  of  the  curve,  as  S  E  =  b  -\-  h'2 ,  >>  b, 

the  corresponding  ordinate  of  the  tangent ;  and  the  other, as  S'  E  —  b h*,  <  b. 

The  cusp  is  therefore  of  the  first  kind. 


D      E 
FIG.  128. 


SINGULAR  POINTS. — CONJUGATE   POINTS.  125 

Ex.  2.  Show  that  y  =  a  f  x  -f-  bx*  -f-  QX*  has  a  cusp  of  the  second 

kind,  if  the  sign  of  x*  be  considered  as  ambiguous,  and  that  the 
equation  of  the  tangent  at  the  cusp  is  y  =  x  -f  a. 

SUG. — To  determine  the  kind  of  cusp,  we  have  — -  = 

26  =b  L£-cx*,  both  of  which  values  are  -\-  for  infinitesimal 
positive  values  of  x.  Therefore  both  branches  of  the 
curve  are  convex  downward  in  the  vicinity  of  the  point, 
and  the  cusp  is  of  the  second  kind.  The  curve  has  the 
general  form  represented  in  the  figure.  There  is  a  point  ' 

of  inflexion  in  the  lower  branch  at  x  =  t,  and  it  cuts 
the  tangent  at  x  =  — . 

Ex.  3.  Show  that  cy*  =  x3  has  a  cusp  of  the  first  kind  at  the  origin. 

Ex.  4.  Show  that   (y  —  b  —  ca;2)2  =  (x  —  a)5  has  a  cusp  of  the 
second  kind  at  (a,  b  -j-  caa). 


FIG.  129. 


CONJUGATE  POINTS. 

186.  DEP. — A  Conjugate  Point  is  an  isolated  point  the  co 
ordinates  of  which  satisfy  the  equation,  while  in  the  vicinity  of  the 
point,  and  on  each  side,  real  values  of  one  co-ordinate  give  imaginary 
values  to  the  other. 

ILL.— In  the  equation  y  =  (a  -f-  x)\/x,  if  x  is  nega 
tive,  y  is,  in  general,  imaginary  ;  but  for  the  particular 
value  x  =  —  a,  y  =  0.  Hence  P  is  a  point  in  the  lo 
cus  ;  and  as  there  are  no  other  points  in  this  plane 
adjacent  to  it,  P  is  an  isolated  or  conjugate  point.  On  ~~P 
the  right  of  the  origin  any  real  value  of  x  gives  two 
real,  numerically  equal  values  to  y,  with  opposite  signs. 
The  curve  has  therefore  two  infinite  branches  on  this 
side,  which  are  symmetrical  with  respect  to  the  axis  PIG  13Q 

of  x. 

Prop. — At  a  conjugate  point  some  one  or  more  of  the  differ- 


-,  -5-^,  etc.,  is  imaginary. 

DEM. — Let  y  =f(x)  be  the  equation  of  a  curve  having  a  conjugate  point  at 
(x,  y}.  Then  letting  h  represent  an  infinitesimal  increment  or  decrement  of  x,  and 
y'  the  corresponding  value  of  y,  we  have  y'=f(x  zfc  K)  =  an  imaginary  quantity,  from 
the  definition  (186).  But 

h' 


y  = 


-- 

-  dx  1 


-f ,  etc. 


126  PROPERTIES  OF  PLANE  LOCI. 

;    Now  as  y  and  h  are  both  real,  to  make  y'  imaginary,  some  one  or  more  of  the 
coefficients  •£,  — |,  — ,  etc.,  must  be  imaginary.     Q.  E.  D. 


188.    Prop.  —  Let     <p(x,  y)  =  u  =  0    be    the     equation    of    a 
curve,  freed  from   radicals;    if  there  is  a  conjugate  point  at  (x,  y), 

the  partial    differential    coefficients    -=-  and  —   are    each    equal   to   0, 

du 
,    civ  dx        0 

and      =  -     =  - 


dni/ 
DEM.  —  Let  —  -^  be  the  first  differential  coefficient  which  is  imaginary  (287). 

Take  the  nth  derived  equation  of  u  =  (p(x,  y)  =  0,  and  we  have  (see  Calculus  1  12], 

—  —  -I  -----------  1  --  =  0,  in  which  the  omitted  terms  are  made  up  of 

dy  dxn  dxn 

differential  coefficients  of  u  with  respect  to  x  and  y,  and  differential  coefficients  of 
y  with  respect  to  x,  of  lower  orders  than  the  nth.  Now,  the  former  are  rational, 
since  u  =  <p(x,  y)  does  not  contain  radicals,  and  differentiating  does  not  introduce 
them  ;  and  the  latter  are  rational  by  hypothesis.  Hence,  in  order  that  the  first 

member  of  the  derived  equation  may  be  0  (which  is  a  rational  quantity),  —  must 

=  0  and  thus  destroy  the  imaginary  factor  —.       Again,    —  -f-  —  •  —  =  0 

du 

du       n  du  .  dy  dx       0 

(Calculus    -)  ;  whence  as  -  =  0,  ~  =  0,  and  -  =  -^  =  -.     o,  E.  D. 

dy 


180.  I*rob. — To  examine  a  curve  for  conjugate  points. 

SOLUTION. — Since  at  a  conjugate  point  —  =  0,  and  -,-  =  0,  if  we  find  the  values 

dx  dy 

of  x  and  y  which  satisfy  these  equations,  these  values  make  known  the  points  to 
be  examined  ;  i.  e.,  they  are  the  critical  values,  the  same  as  in  the  case  of  multiple 

points.     Having  determined  the  critical  values,  we  may  form  ~,  '— -,  -3L  etc.  ; 

dx  dx*   dx3 

and,  if  for  the  co-ordinates  of  any  point  under  consideration,  any  one  of  these 
coefficients  becomes  imaginary,  that  point  is  a  conjugate  point. 


Sen. — The  labor  of  producing  the  higher  orders  of  differential  coefficients  is 
often  so  great,  that  it  is  better,  if  -'-  does  not  become  imaginary,  to  examine 
the  point  by  substituting  successively  a  -f-  h  and  a  —  h  for  a?  in  the  equation 


SINGULAR  POINTS. — SHOOTING   POINTS.  127 

of  the  curve,  a  being  the  value  of  x  to  be  tested,  and  h  an  infinitesimal. 
If  both  values  of  y  found  in  this  way  are  imaginary,  the  point  is  a  conju 
gate  point. 

Ex.  1.  Examine  ay* — #3-{-4ax5 — 5a2#-f  2a3=0  for  conjugate  points. 
SOLUTION  .    —  =  —  3# 2  +  8aa;  —  5a2  =  0,  and  —  =  2ay  =  0,  give  x  =  a,  y  =  0, 

and  x  =  fa,  y  =  0.     Only  the  first  two  of  these  values  (a,  0)  satisfy  the  equation  of 

du 

the  curve  ;  hence  this  point  is  to  be  examined.     To  do  this  we  form  --  =  —  —  = 

dy 

3x2  —  Sax  4-  5a2  . 

.     To  evaluate  this  for  x  —  a,  y  =  0,  we  have 

2uy 

du        Qxdx  —  Sadx        —  2«  dx 

JL  — =       —  __  for  x  —  a,  y  —  0. 

dx,  xady  Aa    dy 

Whence  —  =  —  1,  or  —  =  \/ —  1 .     As  this  is  an  imaginary  quantity,  x  =  a, 
dx*  dx 

y  =  0  is  a  conjugate  point. 

Ex.  2.   Examine  t/2  =  x(x  -f  a)2  for  conjugate  points. 

There  is  a  conjugate  point  at  x  =  —  a,  y  =  0. 

Ex.  3.   Examine  x*  —  ax*y  —  axif  -f-  a2*/2  =  0  for  conjugate  points. 

There,  is  a  conjugate  point  at  (0,  0). 

Ex.  4.  Examine  (c*y  —  x3)*—  (x  —  a)'*(x  —  6)6for  conjugate  points, 
a  being  greater  than  6. 

SUG'S.  — There  is  a  conjugate  point  at  x  =  b,  y  =  —*     Neither  ~,  nor  — ^  are  im 
aginary  for  these  values,  though  -^~  is.     The  better  way  to  solve  this,  is  to  find 

the  critical  values  x  =  a,  y  =  - ,  and  x  =  b,  y=—,  as  usual.      Then  substituting 

in  the  equation  of  the  curve,  we  find  that  both  points  satisfy  the  equation,  and 
hence  are  to  be  examined.  Then  substitute  in  the  equation,  solved  for  y,  the 
values  a  -f-  h  and  a  —  h.  These  give  real  values  for  y  on  one  side  of  the  point  and 

ct* 
imaginary  values  on  the  other.     Hence  x  =  a,  y  =  —  is  not  a  conjugate  point.     In 

the  same  way  substitute  in  the  value  of  y>  b  ±  h,  and  y  is  found  to  be  imaginary 
on  both  sides  of  the  point 


SHOOTING  POINTS. 

190.  DEF. — A  Shooting  Point  is  a  point  at  which  two  or 
more  branches  of  a  curve  terminate,  while  each  branch  has  a  differ 
ent  tangent  a.t  the  point. 

[NOTE.  — This  subject  is  not  of  sufficient  importance  to  justify  an  extended  discussion.    We  shall 
merely  give  a  couple  of  examples.] 


128 


PROPERTIES   OF  PLANE  LOCI. 


Ex.  1.  To  show  that  y  =  x  tan  i-  =  x  cot"1^,  has  a  shooting  point 

OC 


at  the  origin,  if  we  limit  the  discussion  to  x<^  it. 

SOLUTION.  — For  x  —  0,  we  have  y  =  0  -  cot-1!) 
=  0  .  in  =  0  ;  hence  the  curve  has  a  point  in  the 
origin.  In  the  vicinity  of  the  origin,  i.  e.,  for 
very  small  values  of  x,  x  and  cot— lx  have  the 
same  sign,  both  being  -f  on  the  right  of  the 
origin,  and  both  —  on  the  left ;  therefore  y  is  -f- 
near  the  origin  and  the  curve  lies  above  the  axis. 

2 
U  + 


Moreover,    — —  =  — 
dx* 


wherefore     the 


FIG.  131. 


branches  on  both  sides  of  the  origin  are  concave  towards  the  axis  of  x,  and  there 
is  a  salient  point  at  the  origin,  as  in  the  figure.  To  show  that  there  are  two  tan 
gents  to  the  curve  at  this  point,  and  hence  that  it  is  not  a  cusp,  to  which  it  bears 


some  resemblance,  we  form  —  =  cot—  lx  __ 

dx  1  + 

and  for  x  =  —  0,  is  —  fat. 


TLis  foraj=  -f  0  is  -f 


Ex.  2.  To  show  that  y  =  -    --  has  a  shooting  point  at  (0,  0). 

1  +e* 

SUG'S. — For  x  small  and  -f,  y  is  -f  ;   and  for  x 
small  and  — ,  y  is  —  ;  hence  the  branches  lie  as  in 

i 

the  figure.     Again  ^=        — .  -j —r,  which 

i-f  e-  .  a?(i  +  <£)* 


for  x  =  -f  0,  gives  -     =  0  ;  and  for  x  =  —  0,  gives' 


^  —  1.     Therefore  there  is  a  shooting  point  at  (0,  0). 


FIG.  132. 


STOP   POINTS. 

191.  DEF. — A  Stop  Point  is  a  point  at  which  a  single  branch 
of  a  curve  terminates. 

Ex.  1.  To  show  that  y  =  x\ogx  has  a  stop  point  at  the  origin. 

SOLUTION.— In  this  curve  for  all  -f  values  of  x  less  than  1,  y  is  —  and  has  but 
one  value  ;  for  x  =  1,  y  =  0  ;  and  for  x  >  1,  y  is  -f-  and  has  but  one  value.  There 
is  therefore  a  single  branch  of  the  curve  extending  to  the  right  from  the  origin, 
below  the  axis  of  a;  from  x  —  0  to  x  =  1,  and  above  the  axis  of  x,  beyond  x  =  1. 
But  for  x  negative,  y  is  impossible.  .  • .  There  is  a  stop  point  at  the  origin. 

i 

Ex.  2.  Show  that  y  =  e~*  has  a  stop  point  at  the  origin. 


TRACING  CURVES.  129 

SECTION  V. 

Tracing  Curves, 

192.  DEF.— Tracing  a  Curve  is  discovering  from  the  equa 
tion  of  the  curve  and  its  derived  functions  the  general  form  and  lead-'  . 
ing  peculiarities  of  the  curve,  and  its  position  with  reference  to  the 
assumed  axes,  so  that  the  mind  can  conceive  the  locus,  or  that  it  may 
be  sketched  without  going  through  the  details  of  substituting  a  series 
of  values,  as  was  done  in  Section  II.,  Chapter  I. 

Sen. — While  it  is  practicable  to  give  certain  general  directions  for  tracing 
curves,  much  has  to  be  to  be  left  to  the  ingenuity  of  the  student,  as  the 
infinite  variety  of  forms  of  equations  renders  different  methods  expedient 
in  different  cases.  Nor  do  we  know  how  to  trace  the  loci  represented  by 
every  form  of  equation :  this  would  be  equivalent  to  solving  equations  of 
all  degrees. 


103.  I*rob. — To  trace  a  plane  curve  given  by  Us  equation  referred 
to  rectangular  axes. 

METHOD  or  SOLUTION.— If  practicable,  put  the  equation  in  the  form  y  =f(x\ 
Notice  where  it  cuts  the  axes.  Observe  the  limits  arid  infinite  branches.  Examine 
infinite  branches  for  asymptotes.  Find  the  direction  of  curvature  between  estab 
lished  or  characteristic  points.  Determine  positions  of  maxima  and  minima  ordi- 
nates.  Sometimes  it  may  be  serviceable  to  ascertain  the  direction  of  the  curve  at 
certain  points,  as  where  it  cuts  the  axes,  by  means  of  its  tangent  at  those  points. 
Notice  the  position  and  character  of  singular  points. 

SCH. — In  giving  the  above  method  of  tracing  curves,  it  is  not  meant  that 
the  processes  there  detailed  are  necessarily  to  be  gone  through  with  in  the 
order  given,  nor  in  fact  that  they  are  all  to  be  applied  in  tracing  the  same 
curve.  These  are  only  means  to  be  used  as  occasion  may  require.  Again, 
while  these  processes  are  general,  and  constitute  what  is  usually  called 
"  tracing  curves, "  there  are  other  methods  better  adapted  to  certain  cases. 
Of  these  we  shall  give,  in  the  sequel,  three  ;  viz.,  one  when  the  equation  can 
be  put  into  the  form  y  =.  q>(x)  ±  i}>(x},  in  which  y  =  cp(x)  is  a  diametral  locus 
to  that  represented  by  the  entire  equation  ;  another  by  transformation 
from  one  set  of  rectilinear  co-ordinates  to  another  ;  and  a  third  by  passing 
from  one  system  of  co-ordinates  to  another,  as  from  rectilinear  to  polar. 
But  we  will  first  attend  to  a  few  examples  by  the  general  method. 

Ex.  1.  Trace  the  curve  y2  =  ax*  +  bafl. 

SOLUTION. — We  have  y  ==  dr  x\/a  -f-  bx.     On  this  we  observe  that  for  x  =  0, 


130  PROPERTIES   OF   PLANE   LOCI. 

y  =  0.      .• .  The  curve  passes  through  the  origin.     For  y  —  0,  x  =  0,   or  —  -. 

.  • .  The  curve  cuts  the  axis  of  x  also  at  —  -. 

b 

For  all  negative  values  of  x  between  0  and  — y,  y  is  real,  but  beyond  — -  in  a 
negative  direction  y  is  imaginary.     .  • .  x  =  —  -  is  the  limit  of  the  curve  in  this 

direction.     But  for  all  positive  values  of  x,  or  for  all  values  of  x^>  —  -,  y  has  two 

numerically  equal,  real  values,  affected  with  opposite  signs.  .  • .  The  curve  is 
symmetrical  with  respect  to  the  axis  of  x,  and  has  two  infinite  branches  extending 
to  the  right. 

dii  a,  4-  $bx  a  .  dy 

-'•!-  =  ±  — —   —-,  which  for  x  =  —  r,  becomes  -f-  =  ±  <x,  and  for  x  =  0, 

fa          Va  +  kc  b  «    . 

-^  =  ±  \/a.  .  • .  At  (  —  r  5  0  )  the  curve  cuts  the  axis  of  x  perpendicularly,  and  at 
dx  \  o  / 

(0,  0)  it  cuts  it  in  two  directions,  viz.,  at  tan— *(  -f-  N/a),  and  tan~ l( —  \/a).  This 
also  shows  that  (0,  0)  is  a  multiple  point,  a  double  point. 

Examining  for  direction  of  curvature,  we  have  -^  =  ±  —  — ^,  which  is 

4 (a  -j-  bx)*f 

±  between  0  and  —  -,  and  ±  between  0,  and  -f-  oo.  .  • .  At  the  left  of  the  origin, 
the  curve  is  concave  towards  the  axis  of  x,  and  at  the  right,  convex. 

We  have  a  maximum  and  a  minimum  ordinate  at  x  =  —  ^,  y  =  ±  —  \/3a,  as 

appears  by  solving  the  equation  ±  — —     —  =  0. 

v/o  +  bx 

It  only  remains  to  examine  the  infinite  branches  for 
asymptotes. 

dx  i&.r2 

.A.  =  x  —  y—  ==  — ; — -—  =  oo,  for  x  =  oo  ;  and 
"dy       a  -f-  |w 


dy         =F  tixe* 

r-"*  •*  *•— 


Therefore  there  are  no  asymptotes. 

From  this  investigation  the  curve  is  readily  conceived  to 
have  the  form  given  in  the  figure,  which  is  constructed 
assuming  a  =  36.  FIG.  133. 

Ex.  2.  Trace  the  curve  t/2  =  a*x\ 

Results.  The  curve  is  symmetrical  with  respect  to  the  axis  of  x ; 
extends  only  to  the  right ;  is  convex  to  the  axis  of  x  ;  has  two 
infinite  branches  ;  has  a  cusp  of  the  first  kind  at  the  origin,  with 
the  axis  of  x  for  the  common  tangent ;  and  has  no  asymptote. 

Ex.  3.  Trace  the  curve  y  — . 

I       — i—      'T*^ 

Results.  The  curve  cuts  the  axes  at  the  origin  under  an  angle  of  \rt ; 
has  one  infinite  branch  extending  to  the  right  above  the  axis  of  x, 


TRACING   CURVES. 


131 


and  another  extending  to  the  left  below  this  axis  ;  has  a  maxi 
mum  orclinate  at  x  =  +  1,  and  a  minimum  at  x  =  —  1 ;  has  the 
axis  of  x  as  an  asymptote  to  both  branches  ;  has  points  of  in 
flexion  at  (0,  0),  and  at  x  =  \/3,  and  x  =  —  v/3  ;  between  the 
latter  points  is  concave  towards  the  axis  of  x,  and  beyond  them 
is  convex. 

Ex.  4  Trace  y*  =  a3  —  x*. 
Ex.  5.  Trace  (y  —  &)*  =  &. 

Ex.  6.  Trace  at/2  —  x*  +  bx*  =  0. 

Results.  The  curve  cuts  the  axis  of  x  at  right 
angles  at  (b,  0)  ;  has  a  conjugate  point 
at  the  origin ;  has  points  of  inflexion  at 
x  =  ^b  ;  is  concave  to  the  a,xis  of  x  from 
x  =  b  to  x  =  |&,  and  convex  beyond  ;  has 
two  infinite  symmetrical  branches  with 
out  asymptotes. 


FIG.  134 


Ex.  7.  Trace  ay2  —  &  +  (&  —  <?)#2  +  ten 
=  0. 

The  form  of  the  curve  is  given  in  the 
figure.  Observe  that  when  c  —  0  this  locus 
becomes  identical  with  the  preceding,  which 
is  sometimes  called  the  campanulate  (bell 
shaped)  parabola. 


Ex.  8.  Trace  the  Folium  of  Des  Car 
tes,  whose  equation  is  y3  —  Saxy  -f 


Ex.  9.  Trace  y3  =  2ax* 


Ex.  10.  Trace  v2  =  — ' — •     Examine 
x  —  a 

the  curve  for  asymptotes,  for  maxima 
and  minima  ordinates,  for  cusps,  for 
direction  of  curvature,  and  points  of 
inflexion. 


FIG.  135. 


FIG.  136. 


132 


PKOPERTIES   OF   PLANE   LOCI. 


Ex.  11.  Trace  y*  = 


Examine 


the  curve  for  asymptotes,  for  limits,  and 
for  maxima  and  minima  ordinates. 


194.  Prob.  —  To  trace  a  curve  of  the 

second  order,  that  is,  the  locus  of  Ay2  -f 
Bxy  +  Cxs  +  Dy  +  Ex  +  F  =  0,  by  direct 
inspection  of  its  equation. 

SOLUTION.—  One  method  of  solving  this  problem  has  been  given  on  pages  16  19. 
The  present  metbo  1  is  given  as  a  good  algebraic  exercise,  and  in  illustration  of 
the  remark  in  the  preceding  scholium  upon  equations  which  take  the  form 

y  =  <p'^x)  ±  ip^x\ 

Solving  the  equation  for  y  we  have 


±  —  y  (^  —  4A  G)x* 


-f  (D*  — 


FIG.  138. 


1st.  If  we  construct  the  straight  line  of  which 

1 

y  —   —  5k|v»aj  +  D)  is  the  equation  (let  it  be  rep 
resented  by   M  N   in  the  figure),  any  value  of  x 
(as  AD)  which  locates  a  point  (as  P)  in  this  line,   v\ 
locates,  in  general,  two   points  (P',    P")  in  the   ' 
curve,  on  opposite  sides  of  the  line  and  equally 
distant  from  it,  this  distance  being  the  radical  part 

of  the  value  of  ?/.     Therefore  y  = (Bx  -}-  D), 

is  a  diameter  of  the  locus. 

2nd.  For  such  value  or  values  of  a;  as  render  the 
radical  0,  y  has  but  one  value,  and  at  this  point,  or  these  points,  the  locus  cuts  its 
diameter.  Hence  (B*  —  ±AC)x*  -f  2(BD  —  2AE)x  -f  (D*  —  ±AF)  =  0  deter 
mines  where  the  locus  cuts  the  diameter  y  =  —  ^-(Bx  -f  D).  In  general,  this 

gives  two  values  of  x,  indicating  that  the  locus  cuts  its  diameter  in  two  points,  as 
in  the  ellipse  and  hyperbola.  But  if  &  —  4AC  =  0,  the  equation  becomes 
2(BD  —  2AE)x  -f-  (D«  —  4AF)  =  0,  which  gives  only  one  point  of  intersection, 
as  in  the  parabola,  a  result  which  agrees  with  the  fact  that  B*  —  4AC  characterizes 
a  parabola  (62).  Locating  the  point,  or  points,  at  which  the  curve  cuts  its  diam 
eter,  we  know,  if  there  are  two  points,  and  the  curve  is  an  ellipse,  that  it  lies  be- 
tween  these  limits,  or,  if  an  hyperbola,  beyond.  These  facts  will  readily  appear 
by  observing  whether  intermediate  values  of  x  give  real  or  imaginary  values  to  y. 
Thus  the  limits  of  the  curve  appear. 

3rd.  If  the  locus  is  an  ellipse,  the  values  of  y  midway  between  the  two  values  of 


TRACING   CURVES.  133 

x  which  correspond  to  the  extremities  of  the  diameter,  make  known  a  diameter 
parallel  to  tangents  at  the  extremities  of  the  former,  and  hence  determine  the  cir 
cumscribed  parallelogram.  Thus  the  situation  of  the  ellipse  becomes  known. 

4th.  If  the  locus  is  an  hyperbola,  we  can  determine  a  few  values  of  y  corres 
ponding  to  values  of  x  without  the  limits,  and  thus  locate  the  curve.  It  is  often 
expedient  to  find  the  intersections  with  the  axes.  .  t 

5th.  If  the  locus  is  a  parabola,  having  determined  its  diameter  and  vertex,  a  few 
values  of  x  will  make  known  sufficient  points  to  enable  us  to  sketch  the  curve. 
The  intersections  with  the  axes  may  also  be  of  service. 


Ex.  1.  Trace  the  curve  whose  equation  is 
-  2#»  +  2y  -f-  a;  +  3  =  0. 


SUG'S.—  Since  &  —  ±AC  <  0,  the  locus  is  an  el 
lipse.     Solving  for  y,  we  have 


y  =  x  —  1  dr  v/ —  ic*  —  3x  — •  2  ; 
whence  y  =  x  —  1  is  a  diameter,  which  we  construct. 


\/ — .»-  —  3.«  —  2=0,  gives  x  =  —  1,  and  —  2,  the 
limits  of  the  curve.  Between  these  limits  y  is  real, 
and  without  them  it  is  imaginary.  For  x  =  —  1£, 
y  =  —  2,  and  —  3.  Thus  we  find  the  circumscribed 
parallelogram. 


D"    Dr     D 


X 


FIG.  139. 


Ex.  2.  Trace  the  curve  whose  equation  is  t/2  -f  2xy 

x  +  10  =  0. 


2#2  —  4    — 


SUG'S.— As  J3*  —  4: AC  >  0  the  locus  is  an  hy 
perbola.  y  =  —  £c4-2±\/3(x2—  x  —  2).  y  =  —  x 
4-  2  locates  N  M.  From  3(cc'2  —  x  —  2)  =  0,  we 
find  P  and  P'",  at  x  =  2,  and  —  1.  Between 
these  values  y  is  imaginary  ;  hence  the  locus  lies 
beyond  these  points  to  the  right  and  left.  Put 
ting  y  =  0,  we  have  —  2x2  —  x  -f  10  =  0,  whence 
x  =  2,  and  —  2|,  and  the  curve  cuts  the  axis  of 
x  at  C  and  B.  For  x  =  4,  y  =  3  •  5  and  —  7-5 
nearly,  and  we  have  1  and  2.  In  like  manner 
as  many  points  as  we  wish  may  be  found  ;  but 
with  the  diameter  and  intersections  with  the 
axis,  little  or  nothing  more  is  necessary  in  order 
to  form  a  pretty  definite  idea  of  the  situation  of 
the  curve. 


\ 


FIG.  140. 


Ex.  3.  Trace  the  locus  y2  —  Ixy  +  x»  —  4y  -f  x  -f  4  =  0. 


134 


PROPERTIES   OF  PLANE  LOCI. 


SUG'S. — Since  B*  —  ±AC  =  0,  the  locus  is  a  parab 
ola,  y  =  x  4-  2  is  the  equation  of  a  diameter.  For 
x  =  0,  y  =  2.  For  a;  negative,  y  is  imaginary.  For 
a;  =  3,  y  =  8,  and  2. 

Ex's.  4  to  7.  In  like  manner  trace  the  fol 
lowing  :  y2  +  2#j/  4-  3#2  —  4#  =  0  ;  y2  — 
2;ri/  4-  2#2  —  2#  =  0  ;  y2  4- 
4  =  0  ;  and  y5  — 
3  =  0. 


FIG.  141. 


.  frob. — To  tfrace  a  locus  of  the  second  order  by  means  of 
transformation  of  co-ordinates. 

SOLUTION. — We  will  illustrate  this  method  by  an  example.  The  method  itself  is 
altogether  too  tedious  for  practical  purposes,  but  is  highly  important  as  giving  a 
clear  view  of  a  process  which  we  have  occasion  to  use  for  other  purposes.  Let  us 
trace  the  locus  whose  equation  is  y2  -j-  2xy  +  3z2  —  4x  =  0. 

This  is  an  ellipse,  since  B2  —  4J.C<  0.  We  will  find  its  equation  when  referred 
to  its  own  axes.  This  requires  transformation  from  one  rectangular  system  to 
another.  The  formula;  for  this  transformation  are  x  =  xt  cos  a  —  yi  sin  a  -j-  m, 
and  y  =  xl  sin  a  -f-  2/i  cos  a  -}-  n.  Substituting  these  in  the  equation,  we  have 


cos2  a 

— 2  sin  a  cos  a 
4-3sin«a 


—          2  sin'-'a: 


—6  sin  a  cos  a; 


-f-       3cos2a 


Xi24~2ncos  a  7/t4-2nsin  a 
-Jr2?ncoso: 
— 2n  sin  a 
— 6msin  a 
4-  4sina 


-|-2mn 
4- 3m2 


4-2n  cost* 
4-6mcosa 
—  4cos« 

(Eq.  A.) 

As  the  required  form  of  the  equation  is  Ay*  -\-  Bx*  4-  F=  0,  we  desire  to  elimi 
nate  the  terms  containing  xlyl,  and  yl  and  x}.  To  find  the  direction  of  the  new 
axes,  i.  e,  to  determine  the  value  of  a,  and  to  find  the  position  of  the  new  origin, 
i.e.,  to  determine  the  values  of  m  and  n,  which  will  effect  this  reduction,  we  place 
the  coefficients  of  the  terms  to  be  eliminated  each  equal  to  0,  and  solve  the  result 
ing  equations.  These  equations  are 

(1)  2  sin  a  cos  a  —  2  sin2  a  -\-  2  cos9  a  —  6  sin  a:  cos  oc.  =  0  ; 

(2)  2n  cos  a  -{-  2m  cos  a  —  2n  sin  a  —  6m  sin  a  4-  4  sin  a  =  0  ; 

(3)  2n  sin  a  +  2m  sin  a  4-  2n  cos  a  4~  6m  cos  a  —  4  cos  a  =  0. 

From  (1)  we  find  sin  a.  =  .92388,  and  cos  a  =  —  .38268,  whence  a  =  112°  30'. 
m  4-  n  «...  m  4- n 


From  (2)  we  have  tan  a  = 

_  2 


or  —2.414  = 


and  from 


(3),  2.414  = 


n  -f-  3m  —  2'  n  +  3m  —  2  ' 

Solving  these  equations  we  find  m  =  1,  and  n  =  —  1. 


m  -j-  n 

Substituting  these  values  of  sin  a ,  cos  a,  m,  and  n  in  (Eq.  A. ),  we  have,  after 
reduction, 

3.41%!*  -f  .58580?,*  —  2  =  0, 

as  the  equation  of  the  ellipse  referred  to  its  own  axes.     This  gives  the  axes  as 
3.7,  and  1.53. 


TRACING   CURVES. 


135 


To  locate  the  curve  we  have  but  to  construct 
the  new  origin  at  (1,  —  1)  as  AU  and  drawing 
A ,  X ,  making  an  angle  of  112°  30'  with  the  prim- 
itive  axis  of  x,  make  A  [  Y  t  perpendicular  to  it, 
and  on  these  axes  construct  an  ellipse  whose  axes 
are  3.7,  and  1.53. 


Ex.  Trace  by  means  of  transformation 
of  co-ordinates  the  locus  whose  equation     Yx 
is  x2  —  Qxy  -f  2/2  —  GJP  +  2y  +  5  =  0. 

Results.   The  new  origin   (the  centre  of 

the  hyperbola)  is  at  (0,  —  1).      The  FIG.  142. 

transverse  axis,  which  is  the  new  axis  of  x  makes  an  angle  of 
135°  with  the  primitive  ;  and  the  transformed  equation  is  2?/J  — 
4^  —  4  =  0. 


196.  Prob* — To  trace  a  Polar  curve. 

METHOD  OF  SOLUTION.— 1st.  Assign  such  values  to  0  as  give  easily  determined 
values  of  r  :  these  will  usually  be  such  as  0,  fa,  n,  \it,  Zrt,  etc.  ;  or,  if  some  mul 
tiple  of  0  is  involved  in  the  equation,  like  parts  of  these  values.  Thus  if  sin  20 
is  involved,  making  0  =  0°,  sin  20  =  0  ;  if  0  =  15°,  sin  20  =  £  ;  if  0  =  45°, 
sin  20  —  1,  etc.  Construct  these  points.  This  will  often  be  sufficient  to  determine 
the  locus. 

dr 

dO 


ing  functions  of  each  other  and  when  decreasing. 


When 

dQ 


0  the  point  is  an 


apsis,  i  e.  one  at  which  the  curve  is  at  right  angles  to  the  radius  vector  :  at  such 
a  point  r  is  a  maximum  or  minimum.  Thus  in  the  ellipse  when  the  pole  is  taken 
at  the  focus  the  vertices  of  the  transverse  axis  are  apsides. 

3rd.  Examine  the  curve  for  asymptotes,  direction  of  curvature, points  of  inflexion, 
and  any  other  peculiarities  which  may  be  suggested  at  this  stage  of  the  proceeding. 

Ex.  1.  Trace  the  lituus  r  =  — . 

fl* 

SOLUTION. — The  unit  angle 
being  that  whose  arc  equals  ra 
dius  is  about  57°.  3.  Now  let 
ting  a  =  1,  and  0  =  1,  2,  3,  4, 
5,  and  6,  successively,  we  get 
r=±l,  ±.7,  =h  .58,  ±  .5, 
=b  .45,  ±  .41,  ±  .4,  etc.  Lo 
cating  the  positive  values,  we 
get  the  points  1,  2,  3 ,  etc. ;  and 

locating  the  negative  values  we  hare  —1,  —2,  —3,  etc.  The  two  branches  are 
symmetrically  equal. 


136 


PROPERTIES   OF  PLANE  LOCI. 


dr  r5 

Again  -j^=  —  tj>  o.  being  =  1  ;  whence  it  appears  that  r  and  0  are  decreasing 

functions  of  each  other  throughout  all  their  values,  and  the  curve  makes  an  infinite 
number  of  revolutions  around  the  pole,  commencing  from  .00  when  0  =  0,  and 

dr 


reaching  the  pole  when  0  =  oo. 


—  =  —  -  =  0,  gives  /  =  0. 


The  curve  cuts  the 


radius  vector  obliquely,  being  parallel  to  it  at  <x,  and  approaching  perpendicular 
ity  as  r  approaches  0,  or  0  approaches  oo.     The  pole  is  an  apsis. 

»      Since  for  0  =  0,   r  —  oo,   the  subtangent  —  -  =  --  ,  is  0  for  Q  =  0,  and  the 

polar  axis  is  an  asymptote.  • 

To  discuss  the  direction  of  curvature,  we  obtain  the  equation  of  the  spiral  in 
terms  of  the  perpendicular  from  the  pole  upon  the  tangent.     This  equation  is 


whence  -y-  =  --  -  .     There  is  a  point  of  inflexion  at  r  =  ±  \/2, 
)2 


0  =  £,   B  and  B'.     From  B  to  the  right  this  branch  is  convex  toward  the  pole  ; 
and  from  B  toward  the  left  it  is  concave,  as  appears  from  considering  the  sign  of 

-p  for  r  >  >/2,  and  for  r  <  v/2. 

Ex.  2.  Trace  the  locus  whose  polar  equation  is  r  =  a  sin  30. 

SOLUTION.—  If  0  =  0°,  30°, 
60°,  90°,  120°,  1500,  1800,  suc. 
cessively,  r  =  0,  a,  0,  —  a,  0, 
a,  0. 

—  =  3a  cos  30,  which  is  positive 

from  0  =  0,  to  0  ==  300,  negative 

from  0  =  30°  to  0  =  90°,  positive 

from  0  =  90o  to  0  =  150°,  etc. 

Whence  we  see  that  r  begins  at 

0  when   0  =  0°,   increases  till 

0  =  30°,  diminishes  as  0  passes 

from  30°  to  60°,  becomes  0°  for 

0  =  60°,    continues  to  dimin 

ish    (becoming    negative)    as    0 

passes  to  9Qo,  becomes  —a,  at  FIG.  144. 

90°,  etc.     [The  pupil  should  trace  r  through  an  entire  revolution,  in  both  positive 

and  negative  directions.] 

^  =3a  cos  30  =  0,  gives  apsides  at  0  =  30°,  90°,  and  150°,  i.  e.  at  B,  C,  and 
D,  in  the  figure. 
As  r  never  =  oo,  there  is  no  asymptote. 

The  equation  in  terms  of  the  perpendicular  upon  the  pole  is  p  =  ' 

dp        ISa^r  —  8r3 
wnence  —  ==  --  __ 


and  the  curve  is  always  concave  toward  the  pole. 


KATE   OF   CUKVATUKE. 


137 


Ex.  3.   Construct  the  locus  whose  equation  is  x4  —  ax*y 
by  first  passing  to  the  polar  equation. 

SOLUTION.—  The  polar  equation  with  the  pole 


=  0, 


at  the  origin  is  r  = 


SU10 

cos4  0 


(cos«  0  —  sin*  0). 


From  0  =  0°  to  0  =  45°  r  is  real,  finite  and 
passes  from  0  to  0.     Therefore  there  is  a  loop 


in  the  first  octant. 
1  —  3  sin2  fl  —  2  sin"  0 


Letting  a  =  !,  *  = 
=  0,  gives  a  maximum 


M        T       M' 
FIG.  145. 


radius  vector  for  0  =  32°  nearly,  r  =  .45. 

From  0  =  45°  to  0  =  135°,  r  is  negative.  We  will  first  examine  the  values  of  r 
between  0  =  45°  and  90°.  This  gives  a  continuous  curve  in  the  6th  octant,  A3  M. 
Putting  the  equation  in  the  form  r  =  sec  0(tan  0  —  tan3  0),  we  observe  that  as 
tan  0  ^>  1  from  0  =  45°  to  Q  =  90°,  r  rapidly  increases,  and  becomes  —  GO  at  0  =  90°. 
The  branch  in  this  octant  is,  therefore,  infinite.  To  ascertain  more  fully  the  char 


acter  of  this  branch,  we  form  the  subtangent 
cos50 '  tan20  sec  0(1  — 


Bnbt  =  --p  = 

dr 


siii*0(coss0—  sin*0)» 


X 


cos'0 

r -.     Now  since  between  45°  and  90°, 

1  -  -  3  sm20  —  2  sm40          1—3  sin'-2©  —  2  sm*0 

sin20  is  between  \  and  1,  tan  0  between  1  and  GO,  sec20  between  2  and  GO,  and  tan  0 
and  sec  0  increase  much  more  rapidly  than  sin  0,  it  is  easy  to  see  that  subt.  con 
stantly  increases  and  becomes  —  GO  at  0  =  90°.  .  • .  This  is  a  parabolic  branch 
and  approaches  to  parallelism  with  AT. 

Finally,  since  r  =/(sin0,  cosO),  and  only  even  powers  of  cos  0  are  involved,  the 
values  of  r  will  be  repeated  in  the  inverse  order  as  0  passes  from  90°  to  180°. 

+ 


Ex.  4.  Trace  the  locus  whose  equation  is  y2  = 

Uj   • 

to  the  polar  equation. 

SUG. — The  polar  equation  with  the  pole  at  the  origin  is  r  = 
(See  Ex.  11,  193.) 


,  by  passing 


cos  0(1  —  2  cos2  0)* 


SECTION  VI. 

Eate  of  Curvature, 

DEF. — TJie  Curvature  of  a  plane  curve  is  its  rate  of 
deviation  from  a  tangent,  and  is  measured  by  the  subtenses  of  indefi-" 
nitely  small  but  equal  arcs. 

ILL. — Let  M  N  and  mn  be  any  two  circles,  AT  and  AT'  tangents,  and  AS 
and  AS'  infinitely  small  but  equal  arcs.     Then  will  TS  and  T  S  ,  drawn  per- 


138 


PROPERTIES  OF  PLANE  LOCI. 


pendicular  to  the  tangents,  be  the  subtenses  which  measure  the  curvature  of  the 
arcs  of  the  respective  circles  ;  and  we  shall  have  curvature  of  M  N  :  curvature  of 
mn  : :  "TS   :  T'S'.     That  curve  is  said  to  have  the  greatest  curvature  which  de 
viates  most  rapidly  from  its  tangent ;  thus,  in  circles,  A       r_ 
the  greater  the  radius  the  less  the  curvature  ;  L  e.,  the 
curvature  and   radius  are  inverse   functions  of  each 
other.     It  is  also  evident  that  the   circumference  of 
the  same  circle  has  the  same  curvature  at  all  points  ; 
while  in  other  curves,  as  the  conic  sections,  the  curva 
ture  varies  at  every  successive  point.     In  the  ellipse 
the  curvature  varies  from  its  maximum  at  the  extrem 
ities  of  the  transverse  axis  to  its  minimum  at  the  ex 
tremities  of  the  conjugate  axis.     In  the  parabola  and                  FIG.  146. 
hyperbola  the  curvature  is  greatest  at  the  vertex  and  diminishes  as  the  point 
recedes,  becoming  0  at  infinity. 

It  is  the  object  of  this  section  to  present  a  method  of  measuring  curvature,  and 
of  comparing  the  rates  of  curvature  of  the  same  curve  at  different  points,  and  to 
ascertain  the  law  of  variation.  For  this  purpose  a  circle  is  used,  called  the  oscu- 
latory  circle. 

198.  DBF. — An  Osculatory  Circle  is  a  circle  which  has  the 
same  curvature  as  a  given  curve  at  a  given  point ;   or,  it  may  be  de 
fined  as  the  circle  which  has  the  closest  contact  with  a  given  curve  at 
a  given  point. 

ILL. — Let    BDEC   be  an 

ellipse.  If  with  the  centres 
upon  DC  various  circumfer 
ences  be  passed  through  D,  it 
is  evident  that  they  will  coin 
cide  in  very  different  degrees 
with  the  ellipse.  Some  will 
fall  within,  and  others  without. 
Now  the  one  which  coincides 
most  nearly,  as  in  this  case 
M  N ,  is  the  osculatory  circle 
of  the  ellipse  at  the  point  D. 
The  arc  of  the  osculatory  cir 
cle  in  this  case  is  exterior  to 
the  ellipse.  The  osculatory 

circle  at  the  vertex,    as  m"n"  is    within,  and  at  any  other  point,  as  P,  cuts  the 
ellipse,  as  will  be  shown  hereafter. 

199.  DEv.—The  Radius  of  Curvature  is  the  radius  of  the 
osculatory  circle ;   The  Centre  of  Ciirvature  is  the  centre  of 
the  osculatory  circle  ;  and  the  point  of  closest  contact  is  the  point  of 
osculation. 


FIG.  147. 


BATE   OF   CURVATURE. 


139 


200.  DEF.— Contact.     Let  M  N 

and  MM'  be  two  curves  whose  equa 
tions  are  respectively  y  =/(#)  and 
y'  =  <p(x').  Suppose  the  curves  to 
have  a  common  point  P,  so  that  for 
x  =  x'=  AD,y  =  y'=  PD.  Now 
if  x  and  x'  take  the  infinitesimal  in 
crement  D  D ',  which  we  will  represent 
by  h,  designating  the  corresponding  FIG  148. 

values  of  y  and  t/',  by  Y  and  Y'  (S  D;  and  Sf  D')>  we  nave 


and  T= 


-f,  etc. ; 


+' 


Subtracting  the  second  of  these  equations  from  the  first,  we  have 


T  -  T  =  <,-,) 

Now  the  contact  of  these  curves  will  evidently  be  closer  as  Y  -  -  Y' 
(SS;)  is  less.  We  may  therefore  notice  the  following  degrees  of  con 
formity  : 

1st.  If  in  the  case  of  any  two  loci  whose  equations  y  =f(x)  and 
y'  =  <p(x'\  there  is  no  value  of  x  =  x'  which  renders  y  =  y',  there  is 
no  common  point. 

2nd.  If  for  x  =  x',  y  —  y1  and  the  differential  co-efficients  are  un 
equal,  the  contact  is  the  slightest,  and  is  mere  Intersection,* 

3rd.  If  in  addition  to  y  —  y1,  -^  =  —,  and  the  succeeding  coeffi 

cients  are  unequal,  the  contact  is  closer  than  before  and  is  called 
Tangency.    This  is  called  contact  of  the  First  Order. 

dy       dy'        ,  d*y       d*y' 
4th.  If  we  have  at  the  same  time  y  —  y,     ~=      /»  and         ==        ' 


and  the  succeeding  coefficients  unequal,  the  contact  is  of  the  Second 
Order,  etc. 

2O1.  Sen.— A  geometrical  elucidation  of  this  subject  is  obtained  by  con 
sidering  that  "an  infinitesimal  element  of  the  curve  commencing  from  a 
given  point,  being  straight,  is  coincident  with  the  tangent  line  at  that  point ; 
and  the  next  element  of  the  curve,  being  inclined  at  an  angle  to  the  former 
one,  deviates  from  the  tangent.  Now  let  the  two  consecutive  elements 
be  of  equal  lengths,  and  from  the  extremity  of  the  second  let  a  perpen 
dicular  be  drawn  to  the  tangent :  as  this  perpendicular  is  longer  or  shorter, 
the"  curve  will  deviate  more  or  less  from  the  tangent,  that  is,  be  more  or  less 


140  PROPERTIES  OF  PLANE  LOCI. 

bent.  "*  Again,  in  general  a  rectilinear  tangent  is  considered  as  having  two 
points  in  common  with  a  curve,  and  a  circte  three,  since  through  the  three 
consecutive  points  one  circumference,  and  only  one,  can  be  passed. 

202.  DEF. — A  Parameter,  as  the  term  is  used  in  this  and 
similar  discussions,  is  an  arbitrary  constant  entering  into  an  equation 
of  a  locus,  but  which  is  made  variable  by  hypothesis.  Thus  in  the 
equation  y  =  ax  -}-  b,  a  and  b  are  constants  as  ordinarily  considered, 
that  is  have  the  same  values  throughout  the  same  discussion.  Again, 
they  are  arbitrary  constants,  since  they  may  have  any  values.  Finally, 
we  may  consider  how  a  straight  line  changes  position  when  a  and  6 
vary  continuously.  In  this  case  a  and  6  are  called  parameters. 


203.  Prop. — If  one  curve  be  given  in  species,  magnitude,  and 
position,  ilial  is  entirely  given,  and  a  second  given  only  in  species,  in  gen 
eral  the  highest  order  of  contact  possible  is  equal  to  the  number  of  para 
meters  in  the  equation  of  the  second  curve  less  one. 

ILL. — As  this  proposition  usually  seems  to  the  learner  quite  abstract,  we  will 
give  a  familiar  illustration  of  its  meaning  before  proceeding  to  its  demonstration. 
Let  9y2  -f-  4#2  =  36  be  the  first  locus.  The  species  is  ellipse ;  the  magnitude  is 
determined  by  the  value  of  the  axes  6  and  4  :  the  form  of  the  equation  determines 
the  position  of  the  locus.  Thus  this  curve  is  given  in  species,  magnitude  and 
position,  or  entirely  given.  Constructing  it  we  have  the  ellipse  in  the  figure. 

Let  the  second  equation  be  that  of  a  circle  in  its  general  form, 
viz.,  (x  —  ra)2  -\-(y  —  w)2  =  r2,  in  which  ra.  n,  and  r,  are  arbitrary 
constants,  which  we  propose  to  treat  as  variables,  thus  making 
them  parameters.  It  is  evident  that  the  closeness  of  contact  of 
these  two  curves  will  depend  on  two  things,  the  value  of  the  ra 
dius,  aud  the  position  of  the  centre  ;  but  the  position  of  the 
centre  depends  upon  the  values  of  m  and  n.  Hence  the  closeness  ' IG' 

of  contact  depends  upon  the  values  of  the  three  parameters  m,  n,  and  r.  Thus  if 
P  be  the  common  point,  by  locating  the  centre  at  C,  and  using  CP  as  radius,  it 
is  evident  that  the  contact  is  much  closer  than  when  C'  is  the  centre  and  C  P  the 
radius.  There  is  therefore  some  position  of  the  centre  and  some  value  of  the  radius 
which  will  give  the  circle  closer  contact  than  any  other.  Moreover  it  is  evident 
that  we  have  given  the  widest  possible  opportunity  for  varying  the  contact,  by  taking 
that  form  of  the  equation  of  the  circle  which  has  the  three  parameters  m,  n,  audr. 
We  will  now  give  the  demonstration. 

DEM. — Let  y  — /(x),  and  y'  =  tp(x'}  be  the  equations  of  the  loci.  In  order  that 
we  may  make  y  —  y'  for  some  value  of  x  =  x  we  must  have  liberty  to  impose  one 
arbitrary  condition  (i.  e.,  to  vary  the  second  locus  in  at  least  one  respect),  but  this 
requires  one  parameter.  If,  in  addition  to  this  parameter,  there  is  a  second  (L  e.,  if 
we  can  vary  the  curve  in  another  respect)  we  can  impose  another  arbitrary  condi- 

*  Price's  Infinitesimal  Calculus. 


KATE   OF   CURVATURE. 

tion,   as   -p  =  — ,  and  so  on  for  any  number  of  parameters.     Hence  we  see  that 
dx       dx' 

one  parameter  makes  intersection  possible  ;  two  make  tangency  or  contact  of  the 
first  order  possible  ',  three  contact  of  the  second  order,  etc. 

204.  COR.  1. —  The  right  line  can  have  in  general  no  higher  order  of 
contact  than  the  first  (tangency),  since  its  equation  y  =  ax  -f  b  has  but 
two  parameters  a  and  b. 

20o.  COR.  2. — As  the  equation  of  the  circle  in  its  general  form  has 
but  three  parameters,  it  can  in  general  have  no  higher  order  of  contact 
than  the  second. 

200  •  COR.  3. — The  parabola  can  have  contact  of  the  third  order,  and 
the  ellipse  and  hyperbola  of  the  fourth. 

2O7.  Sen. — This  discussion  assumes  that  y  =f(x),  which  is  given  in  all 
respects,  is  of  such  a  character  as  to  allow  of  any  degree  of  contact.  Of 
course  the  possibilities  of  contact  are  limited  as  much  by  one  of  the  loci  as 
by  the  other.  Thus,  if  the  first  locus  were  a  circle  and  the  second  an 
ellipse,  the  contact  could  not  in  general  be  above  the  second  order,  although 
.  the  ellipse  has  a  possible  contact  of  the  fourth  order  with  other  curves. 
Again,  in  this  discussion  we  have  said  "in  general,"  since  exceptions  occur 
at  certain  singular  points.  Some  of  these  will  be  noticed  hereafter.  Thus 
far  we  have  given  the  broader  view  of  osculation,  although  for  the  practical 
purpose  of  the  measurement  of  curvature  we  might  limit  our  view  to  the 
circle,  as  we  shall  do  in  the  following  propositions. 

205.  JP^ob. — To  produce  the  general  differential  formula;  for  the 
value  of  radius  of  curvature  and  the  co-ordinates  of  the  centre  of  curva 
ture  of  any  plane  curve,  in  terms  of  the  co-ordinates  of  the  given  curve. 

SOLUTION  1. — Let  y=f(x)  be  the  equation  of  the  given  locus,  and  (x  — w)2  -f- 
(y'  —  n)2  =  r2  the  equation  of  the  circle.  Now  as  the  equation  of  the  circle  con 
tains  three  arbitrary  constants,  m,  n,  and  r,  we  may  impose  three  conditions  and 
find  the  values  of  these  constants  which  fulfill  them.  The  conditions  requisite  for 

the  closest  contact  which  a  circle  can  have,  are,  for  x  =  x' ,  y  =  y' ,  -j-  =  —,,  and 

—  —  — ~.     These  therefore  are  the  conditions  to  be  imposed,  and  from  which 
dx*       dx* 

the  values  of  TK,  ?i,  and  r  are  to  be  obtained.     In  any  given  case  it  will  be  suffi 
cient  to  find  the  values  of  y,  -?-,  and  -r— ,  in  the   equation  of  the  locus,  and  also 
dx  dx2 

the  values  of  y',  j-;,  and  — --,  in  the  general  equation  of  the  circle,  and  equating 

the  corresponding  values  find  from  the  three  equations  thus  formed  the  values  of 
m,  n  and  r. 

But  for  practical  purposes,  general  formula;  are  more  convenient.  These  are 
readily  produced,  as  follows  : 


14:2  PROPERTIES   OF   PLANE   LOCI. 

Differentiating  the  equation  of  the  circle  twice  in  succession  we  have 

(1)  (x'  —  m)  -f  (y'  —  n)-j^  =  0 

dy 2          »          d'2y'      n 

In  these  equations  and  the  general  Equation  of  the  circle 

(3)  (x'  —  m)*  +  (y'  —  n)2  =  r2, 

we  can  now  substitute  the  values  of  y,  ^-,  and  —^  as  obtained  from  the  equation 
of  the  given  locus  considering  x  =  x',  and  have 
4)  (x  _  m)  +       ._  n)??  =  0 

dtf2  d'}y       _ 

(5)  i  _|_  _±i — 1_  ^  - —  n) — -  =  0, 

(6)  (x  —  m)2  -f  (y  —  n)2  =.r2- 

In  order  to  solve  these  equations  for  r,  m,  and  n,  we  get  from  (5) 

1  ~^~  of?2 

(7)  y  —  n  = --L.,  which  substituted  in  (4)  gives 


(8)    x  —  m  = 


(9)     r=± 


dx2 

I  1  ~l~  ~^  J""1" 
\       ox1  /(la! 


dx* 


.    Substituting  these  values  in  (6)  and  reducing  we  have 


,  which  is  the  formula  for  radius  of  curvature. 


The  co-ordinates  of  the  centre  (m  and  n)  are  written  at  once  from  (8)  and  (7). 
They  are  » 


(10)     m  —  x  — 


(11)     n  = 


,  and 


Q.  E.  D. 


SOLUTION  2. — Let  P,  P',  and  P",  ^uj.  150,  be 
three  consecutive  points  through  which  the 
curve  M  N ,  whose  equation  is  y  =  f(x),  and 
the  osculatory  circle  mn  whose  equation  is 
(x  —  m)2  -f-  (y'  —  n)2  =  r2,  pass,  and  between 
which  they  coincide.  PP',  and  P'P"  are  then 
to  be  considered  straight  lines.  Pass  a  circum 
ference  through  the«e  three  points  by  erecting 
perpendiculars  at  the  middle  points  of  the 
chords  PP',  and  P  P".  These  perpendiculars, 
C  D  and  C  D',  are  consecutive  normals.  Hence 


FIG.  150. 


KATE  OF  CURVATURE. 


143 


the  centre  of  the  oscillatory  circle  may  be  conceived  as  the  intersection  of  two  consec 
utive  normals. 

Having  premised  the  above  fact,  let 
MN,  Fig.  151  be  a  curve  whose  equation 
is  y  =f(x).  Let  PC  and  P'C  be  two  con 
secutive  normals.  Then  is  C  the  centre 
of  osculation,  and  PC  =  r,  the  radius  of 
curvature.  Again,  let  s  represent  the 
length  of  the  curve,  and  ip,  the  angle  at  the 
centre  of  the  osculatory  circle  to  radius 
unity.  As  P  and  P'  are  consecutive  points, 
P  P'  =  ds,  F>\-=dx,P'L.  =  dy,  and  ce  =  FlG-  151. 

dip  are  contemporaneous  infinitesimal  elements  of  s,  x,  y,  and  ip  respectively. 
Moreover,  drawing  EH  parallel  to  P'C,  the  angle  PH  E  =  PCP'  =  dip  is  the 
corresponding  infinitesimal  element,  of  the  angle  which  the  normal  makes  with 

the  axis  of  x  ;  or  dip  =  £tan~*( f-\  as  tan  -'( -^-  )  is  the  angle  a  normal 

\      dy  /  V      dy  ' 

makes  with  the  axis  of  a;.  Now  since  ce  =  dip  is  an  arc  at  a  unit's  distance  from 
the  centre,  and  PP'  =  ds,  is  the  corresponding  arc  at  r  from  the  centre,  we  have 

(1)     ds  —  —  rdip,  or  r  =  —  — ,  the  —  sign   signifying  that  s  is  a   decreasing 
function  of  ip. 

<doc  \ 
j  with  respect  to  x, 

dty  dx 


But  ds  =  ±  v' 


dip=  d  tan-1!  —  —  1  = 

v      dy  ' 


ds  and  dip  in  (1),  we  have  r  =  ± 


dy-  -f 


(dy* 


d*ydx 


Substituting   these  values  of 


•,  as  before. 


dx* 


2O9.  SCH. — Since  the  numerator  of  the  value  of  r,  equals  — ,  and  x  and 
s  are  increasing  functions  of  each  other,  it  is  always  to  be  regarded  as  -f  ; 
whence  we  see  that  the  sign  of  r  depends  upon  the  sign  of  — .  There 
fore  r  is  to  be  considered  +  when  the  curve  is  convex  downward,  and  — 
when  it  is  convex  upward  (169}. 

Ex.  1.  Find  the  radius  of  curvature,  and  the  co-ordinates  of  the 
centre  of  osculation  in  the  common  parabola. 


SOLUTION.— We  have 


dy 


the  sign.     Again  m  =  x  — 


d# 


144 


PROPERTIES   OF   PLANE   LOCI. 


210.  COR.  1.  —  The  radius  of  curvature  at  the  vertex  of  the  common 
parabola  is  half  the  latus-rectum,  since  at  this  point  y  =  0,  and  r  = 


211.  COR.  2.  —  The   radius  of   curvature   in  the   common  parabola 

varies  as  the  cube  of  the  normal,  since  normal  =  (y2  -f  p8)"3",  and  r  = 
(normal)3 


Ex.  2.  What  is  the  radius  of 
curvature  of  a  parabola  whose 
latus-rectum  is  9,  at  x  =  3? 
"What  are  the  co-ordinates  of 
,the  centre  of  curvature  ?  What 
are  they  at  the  vertex?  Con 
struct  such  a  parabola  with  the 
oscillatory  circles  in  position. 

Anxwer.  For  x  =  3,  r  =  C  P 
=  16.04;  m=  AE  =  13£;n  = 
EC  =  —  6.91.  At  the  vertex 


and  n  =  0. 


FIG  152. 


Ex.  3.  Find  the  radius  of  curvature  of  the  ellipse,  and  the  co-ordi 
nates  of  the  centre  of  curvature. 


dy  B»x        ,  d*y 

SUGGESTIONS.    --  =  --  ,  and  —  -  = 


dx 


;   whence  r  = 


=  (C2  being  =^-^);  and  n  =  y- 


212.  COR.  1. — The  radius  of   curvature  at  the  extremities  of  the 

T>j 

transverse  axis  of  an  ellipse  is  half  the  lotus-rectum,  or  —  ;   and  at  the 

A. 

A2 

extremities  of  the  conjugate  axis,  it  is  — . 


213.  COR.  2. — The  radius  of  curvature  in  an  ellipse  varies  as  the  cube 

of  the  normal,  since  normal  =  —  v/A4y2 -f  B4x*,  giving  r  =  ^n°r       ^      . 

Aa  J34 


RATE    OF    CURVATURE.  145 

Ex.  4.  Find  the  radius  of  curvature  at  x  =  2, 
and  also  at  the  vertices  of  the  axes  of  the  ellipse 
whose  axes  are  8  and  4.  Find  also  the  centre  of 
curvature,  and  construct  the  osculatrices. 

Results.  At  a:  =  2(  P),  (.375,  —  3.9),  i.  e.  C,  is  the 
centre  of  curvature  and  r  =  5.86(PC).  At 
the  vertices  of  the  transverse  axis  C  is  the 
centre  of  curvature  and  r  =  1. 

214.  Sen. — The  centre  of  curvature  being  the  intersection  of  two  con 
secutive  normals,  it  is  always  in  the  normal  drawn  to  the  point  of  osculation. 
Hence  having  found  the  value  of  r  in  any  given  case,  if  we  can  draw  the 
normal  geometrically,  it  is  not  necessary  to  find  the  co-ordinates  of  the 
centre  of  curvature  in  order  to  draw  the  osculatrix.  If,  however,  we  do 
not  know  how  to  draw  the  normal  geometrically,  the  co-ordinates  of  the 
centre  of  curvature  give  a  point  in  it,  whence  it  can  be  drawn. 

Ex.  5.  Find  the  radius  of  curvature  of  logarithmic  curve,  a?  =  logy. 

3 

(wz2  4-  7/1)"2" 

r  = '- . 

my 

Ex.  6.  Find  the  radius  of  curvature  in  the  cubical  parabola,  ?/3  =  a?x. 


Ex.  7.  Find  the  radius  of  curvature  of  the  curve  y  —  sfi  —  x1*  -f  1, 
where  it  cuts  the  axis  of  y,  and  also  at  the  point  of  minimum  ordi- 
nate.  How  does  it  appear  from  the  operation  that  the  curve  is  con 
cave  towards  the  axis  of  x  at  the  former  point  and  convex  at  the  lat 
ter?  (See-Ffy97.) 

At  the  first  point  r  =  —  ^  ;  at  the  second  r  =  ±: 

Ex.  8.  Find  the  radius  of  curvature  of  the  locus  t/3  =  6a?*  -f  x*. 
How  does  it  appear  that  this  locus  is  always  concave  towards  the  axis 

/  7/4    J_    ( A.T   4-    7-2 ^-"2" 

of  x ?     (See  Fig.  106. )  r=^— '    (       o      j  '    . 

Ex.  9.  Prove  that  in  the  cycloid  the  radius  of  curvature  equals 
twice  the  normal.  Construct  a  cycloid  and  upon  this  principle  draw 
the  osculatory  circle  at  several  points.  What  is  the  radius  of  curva 
ture  at  the  points  where  the  cycloid  meets  its  base  ?  What  at  the 
vertex  ? 


2 IS.  I*TOp. — At,  a  point  of  inflexion  a  rectilinear  tangent  to  a  curve 
has  contact  of  the  second  order. 


146  PKOPERTIES  OF  PLANE  LOCI. 

DEM. — Let  y  —f(x}  be  the  equation  of  the  curve,  and  y'  =  ax'  -f-  6  be  the 
equation  of  a  right  Hue.     At  a  point  of  tangency  in  general  we  have  for  x  =  x', 

y  =  y',  and  ---  =  -^7.     But  at  a  point  of  inflexion  —  =  0.     Also  in  the  equation 

CIJC         (/X  CuC 

t?^?/'  C?*t/          ci^l/' 

of  the  right  line  — —  =0.     . •.  — -  =  ~:—,  and  we  have  the  conditions  of  contact 
dx'2  dx*       dx'2 

of  the  second  order.     Q.  E.  D. 


210.  IPt'Op.  —  At  points  of  maximum  and  minimum  curvature  of 
any  plane  curve,  the  osculatory  circle  has  contact  of  the  third  order. 


DBM.  —  At  such  points  —  =  0.     Now  differentiating  r  =  -  —  —  -   we  have 


.. 

dx  ~ 


*y      3^  (  ^ 

Whence  -^  =  - 
dx3 


But  in  the  circle  we  have  found  y  —  w  =  —  — -^ .     Differentiating  this,  and 


finding  the  value  of  — ,  we  have  —  =  ,  •-.     Therefore  as  the  third  differ- 

dx3  dx3          1    ,   dy^ 

dx* 

ential  coefficient  is  the  same  in  the  circle  as  at  a  point  of  maximum  or  minimum 
curvature  of  any  plane  curve,  the  contact  is  of  the  third  order  at  such  points. 
Q.  E.  D. 

217.  COB. — The  contact  of  the  osculatory  circles  at  the  vertices  of 
the  conic  sections  is  closer  than  at  other  points,  a  fact  which  is  also  appa 
rent  in  the  construction. 


218.  Prop.  —  When  contact  is  of  an  even  order  the  loci  intersect  ; 
but  when  of  an  odd  order  they  do  not. 

DEM.  —  Let  Y=f(x)  and  y  —  (p(x)  be  the  equations  of  the  two  loci.     Then  the 
difference  of  their  ordinates  corresponding  to  x  ±  h  is  Y'  —  y'  = 


dY      dy\(± 

~dx  dx)  I  T~\de*  d.c*  2 
-f-,  etc.  Now,  when  the  order  of  contact  is  even,  the  first  term  of  this  difference 
which  does  not  reduce  to  0,  and  which  fixes  the  sign  of  the  sum  of  the  series, 
contains  an  odd  power  of  ±  h  ;  and  hence  Y'  —  y'  is  positive  for  -f-  h,  and  nega- 


BATE   OF   CUKVATUKE.  147 

tive  for  —  h,  showing  that  the  loci  intersect  at  the  point.  If,  on  the  other  hand, 
the  contact  is  of  an  odd  order,  the  first  term  which  does  not  reduce  to  0  contains 
an  even  power  of  ±  h  ;  and  hence  does  not  change  sign  with  h,  and  one  of  the 
curves  lies  within  the  other,  as  in  tangency.  Q.  E.  D. 

210.  COR. — The  oscillatory  circle  always  cuts  a  conic  section  except 
at  points  of  maximum  and  minimum  curvature. 


220.  Prob.  —  To  produce  the  formula  for  radius  of  curvature  in 
terms  of  Polar  Co-ordinates. 

SOLUTION.  —  We  will  produce  this  formula  by  transformation  of  co-ordinates,  as 
the  process  affords  both  a  good  exercise  in  transformation,  and  also  in  changing 
the  independent  variable.  In  order  to  distinguish  between  radius  of  curvature 
and  radius  vector,  let  the  former  be  represented  by  R,  and  the  latter  by  r. 


We  have  already  seen  that  R  =  _Jl_-_  (2O8).  But  this  formula  was  pro 
duced  on  the  assumption  that  x  was  equicrescent,  and  hence  d(dx}  =  0.  To  give 
it  the  more  general  form,  we  have  only  to  remember  that  d*y  =  d(  —  )dx  = 
d2y  dx  —  d'2x  dy 


dx 


;  and  hence  that  the  general  formula  is 


d'2y  dx  —  d2x  dy 

The  equations  for  transformation  are  y  =  r  sin  0,  and  x  =  r  cos  0.     Considering 
0  equicrescent  (L  e.  as  the  independent  variable)  and  differentiating,  we  get 
dy  =  dr  sin  0  -f-  r  cos  0  d0, 

di/2  =  dr2  sin2  0  -f-  2r  sin  0  cos  QdrdB  -f-  r2  cos2  0  d02, 
d2y  =  d2r  sin  0  -}-  2  cos  QdrdB  —  r  sin  0  d02, 
d.r  =  dr  cos  0  —  r  sin  0  d0, 

dx'2  =  dr2  cos*  0  —  2r  sin  0  cos  0  dr  d0  -f-  r2  sin2  0  d02, 
and  d2#  =  d2r  cos  0  —  2  sin  0  dr  d0  —  r  cos  0  d02. 

4  £ 

.  • .  (dafl  -f-  dy2r  =  (dr2  -f-  r2d02)  ,  and 


d  y  dx  —  d-x  dy  =  2(sin20  -f  cos20)dr2d0  —  rd0(sin20  -f-  cos30)d2r-f-  (sin20  4-  cos20)r2d08 

=  2dr»  dQ  —  rdQ  d2r  -f  r2  d03. 
Whence,  substituting,  we  have 

7?  —  (dr2  -f-  r2<702)'  (d&  "*" 

~  -f-  r*d0:»  ~~    dr2         d!r         a' 


221.    COR.  —  /S'mce    the    length   of   a  normal  to   a  polar    curve   is 
(—  -f  r2V,  (107],  representing  the  normal  by  N,  we  have 


dr*         dr 


PROPERTIES   OF  PLANE  LOCI. 
Ex.  1.   Find  the  radius  of  curvature  of   the  logarithmic  spiral, 


SOLUTION.     —  =  a  log  a,  and  —  -  =.  a  log2  a. 


—  ra   Iog2a  +  r2         a26  log2  a  -|-  r2 

« 
the  polar  normal.     [The  first  reduction  is  made  by  remembering  that  r  =  a  .  ] 

Ex.  2.  Find  the  radius  of  curvature  of  the  lemniscate  of  BernouilH, 
r*  =  a2  cos  20. 


„  dr  _        a2  sin  26 

~ 


a  sin  20 


and 


««  sin2  20 
cos  20~ 
2a  cos2  20  +  a  sin2  29 


Substituting  these  values,  we  have 


a2  sin2  26 


3r' 


cos  20 


SECTION   VIL 

Evolutes  and  Involutes, 

222.  DEF. — An  Evolute  of  a  curve  is  the  locus  of  the  centre 
of  curvature.     The  primary  curve  is  called  the  Involute. 

ILL. — If  M  N  be  a  plane  curve,  and 
the  centre  of  curvature,  C,  be  deter 
mined  for  any  point,  P  ;  then,  as  P 
passes  along  the  curve  to  P',  P",  P'", 
etc.,  the  centre  of  curvature  will  de-  ' 
scribe  another  curve,  as  C ,  C',  C",  C'", 
etc.  M '  N  being  thus  described  is  the 
e  volute  of  M  N  ;  and  M  N  is  the  inuo- 

luleof  M'N. 

FIG.  154. 


EVOLUTES   AKD   INVOLUTES. 


149 


223.  I*rob. — Given  the  equation  of  a  plane  curve,  to  find  the  equa 
tion  of  its  evolute. 

SOLUTION. — Let  y  =f(x)  be  the  equation  of  the  given  curve,  as  M  N  Pig.  154. 
Now  the  co-ordinates  of  the  centre  of  the  oscillatory  circle  are  the  co-ordinates  of 
the  evolute.  Hence,  if  we  combine  the  equations 


^  <1x* 
d*y    ' 


with  y  —/(#),  and  eUminate  x  and  y  there  will  result  an  equation  between  m  and 
n,  the  co-ordinates  of  the  evolute,  which  is  therefore  its  equation. 

SCH. — The  equation  of  the  locus,  y  =f(x]  is  needed  in  connection  with 
the  values  of  m  and  n,  only  when  these  values  contain  both  x  and  y. 

Ex.  1.  Find  the  equation  of  the  evolute  of  the  common  parabola. 

SOLUTION.  — We  have  —  =  -,  and  -1—  =  —  —.     Whence 
dx       w  dx*  v3 


m  =  x  — 


,    dy*\dy 
r  dx*Jdx 


-f 


dx* 


N    XN' 


and  n  =  y 


Now  from  m  =    y  J~  P  ,  and  n  =  —  -a,  if  we  elim 
inate    y   we    obtain,    after   a    little    reduction,  FIG.  155. 

Q 

n'z  =  - — (m — pY,  which  is  the  equation  sought.      Tracing  the  curve  we  find 
M'A'N',    Fig.   155.      If  we   transfer  the  origin  to  A',   the   equation   becomes 

~~  27pm 

224.  SCH. — This  locus  is  called  the  Semi-cubical  Parabola,  any  curve 
having  infinite  branches  and  no  rectilinear  asymptotes  being  called  a  para 
bola. 

Ex.  2.  Find  the  evolute  of  the  circle. 


SUG'S.  —  The  equations  to  be  solved  are 


r2  y3  x 


n  =  y  -  ~-  =  0, 
y*i* 

and  .T2  -f-  y2  =  r2.     Whence  m  =  0  and  n  =  0,  for  all 

values  of  x  and  y,  and  the  evolute  is  a  point,  the  centre.     This  is  evidently  correct, 
since  all  normals  (radii)  of  the  circle  meet  at  the  centre. 


150  PROPERTIES   OF   PLANE  LOCI. 

Ex.  3.  Find  the  evolute  of  the  ellipse. 


SUG'S.—  We  have  m  =  ~,  n  =  — 


,  and 


=  A*B*,  from  which  to  eliminate  x  and  y  and  find  an 
equation  between  m  and  n,  the  co-ordinates  of  the  evolute. 


33 


J5V 


The   equation   sought  is  A3m3  +  J5         =  (A*  — 
The  evolute  is  of  the  form  CC"C'C'",  Fig.  156. 

Ex.  4.    Produce  the  equation  of  the  evolute  to 
the  cycloid. 


FIG.  156. 


dy      v/2rv  —  V'2       -,  d*y          r 
SuG's.-We  have  JL  =  -  |_*_,  and  ^J  =  -  -.    Whence  m 

and  n=—y.     .•  .  y  =  —  n,  and  x  =  m  —  2  v/—  2rn  —  n*.     Substituting  these  in 
the  equation  of  the  cycloid,  we  have  m  =  vers-^—  n)  -f  \/  —  '2m  —  n2. 


COR.  —  T/ie  evolute  of  a  cycloid  is  an  equal  cycloid. 


DEM.— The  equation 
x  =  vers-1!—  y)  -f  v/—  2n/  —  y*  is 
the  equation  of  a  cycloid  referred  to 
its  highest  point  A,  as  the  origin  and 
having  a  tangent  at  that  point  as  the 
axis  of  abscissas  and  the  axis  of  the 
cycloid  for  the  axis  of  ordinates. 
This  will  readily  appear  by  produc 
ing  the  equation  under  these  conditions.  Thus  in  Fig.  157  AD  —  AE-|-ED  = 
CH  +  FP  =  CB—  HB+Fpt=arcHPE  —  arc  H  P+ FP  =  arcEP-f  FP. 
But  AD  =  x,  PD  =  —  y,  arc  EP  =  y  M 

vers—1  E  F  =  vers-1  P  D   =  ver~ l  ( —  y), 
and 


FIG.  157. 


=  v/EF  X  FH  :=VPD  X  FH 


=  v(—  y)C*r  —  PD)  =  \/( — y)( 

v/ —  *&ry  —  2/2.     Hence  x  =  vers— 1( —  y}  -f- 

Thus  we  see  that  M  Mg.  158  being  a 
cycloid  whose  equation  is  a;  =  vers—  ly —  ^IGt  -^* 

\72ry  —  y-1,  N  is  its  evolute  whose  equation  is  m  =  vers-'( —  n)  +  v/ — 2rn  —  n«, 
referred  to  A  as  its  origin.  This  equation  is  satisfied  for  none  but  negative  values 
of  n,  and  gives  m  =  0,  2itrt  ±itrt  etc.,  for  n  =  0 ;  and  also  for  n  =  —  2r,  m  =  itr, 
3*r,  etc.,  as  it  should. 


EVOLUTES   AND   INVOLUTES. 


151 


Son.  —The  student  will  readily  dis 
cern  tlie  character  of  the  evolute  of 
the  cycloid  from  the  property  that-  the 
radius  of  curvature  is  always  twice 
the  normal.  Thus  if  the  two  circles 
C,  and  C'  roll  along  the  bases  AX 
and  A'X'  at  equal  rates  so  as  to  keep 
their  centres  in  the  same  vertical  line 
P'  will  describe  the  evolute  as  P  does 
the  involute. 


FIG.  159. 


22(>.  Prop. — A  (produced]  normal  to  an  involute  ix  tangent  to  the 
evolute,  the  point  of  tangency  is  the  centre  of  curvature,  and  consequently 
the  nor  ma'  thus  produced  is  the  radius  of  curvature. 

DEM.— Let  (m,  n)  be  any  point  in  the  evolute  of  AM 
from  it  draw  a  normal  to  AM,  and  let  (X,  y}  be  the 
point  at  which  it  is  normal.  The  equation  of  this  nor- 


mal  is 


or 


—  --(x  — 


Now  as 


the  point  (m,  n)  changes  position  (x,  y}  also  changes, 
and  to  observe  the  law  of  change  we  differentiate  (1)  for 
x,  y,  m  and  n  as  variables.  This  gives 

dy*  —  dndy    . 


dx  —  dm  -f 


dm 


But  as  (m,  n)  is  in  the  evolute  we  have  (2O8)  y 


whence 


1  4.  j?£  4.  (T/  _  n)—  =  0.     Therefore  dropping  these  terms,  (2)  becomes 
1    dx2  dx'2 


dm        dn  d; 
~  dx  "'  ~dx*" 

dx 
Hence  the  equation  y  —  n  =  —  -p(# 

the  involute  at  (x,  y},  may  be  written 


=  0,  and =-  =  TT-. 

ay        dm 

(x  —  m),  which  is  the  equation  of  a  normal  to 


dn 


which  is  the  equation  of  a  tangent  to  the  evolute  at  (m,  n).     Q.  E.  D. 

227.  COR. The  radius  of  curvature  and  the  arc  of  the  evolute  vary 

by  equal  increments ;  that  is,  the  arc  of  the  evolute  between  two  centres  of 
curvature  equals  the  difference  between  the  corresponding  radii  of  curva 
ture. 


152 


PROPERTIES   OF  PLANE   LOCI. 


DEM. — Since  the  radius  of  curvature  is  a  tangent  to 
the  evolute  it  coincides  with  the  arc  between  two  con 
secutive  points.  Thus  P  and  P'  being  consecutive 
points  on  the  involute,  the  radius  at  P  is  to  be  consid 
ered  as  having  the  two  consecutive  points  C  and  C' 
common  with  the  evolute  to  which  it  is  tangent ;  and  as 
P  passes  to  P',  the  radius  of  curvature  so  changes  po 
sition  as  to  have  the  consecutive  points  C'  and  C" 
common,  and  to  coincide  with  the  curve  between  them. 
Thus  it  appears  that  the  radius  of  curvature  and  the 
arc  of  the  evolute  vary  by  equal  increments. 


FIG.  161. 


228.  SCH. — From  these  relations  it  is  easy  to  see  how  an  involute  may 
be  described  mechanically  from  its  evolute.     For  example,  to  draw  a  para 
bola,  make  a  pattern  of  the  form  AOCM  Fig.  161,  the  edge  OCM  being 
the  arc  of  an  evolute  to  the  required  parabola,  and  AO  =p,     Fasten  a  cord 
at  M  and,  wrapping  it  around  the  edge  of  the  pattern,  fasten  a  pencil  to  the 
free  end  at  A.     Keeping  the  string  tight,  move  the 

pencil  along  as  from  A  to  P,  P',  R,  and  it  will  de 
scribe  the  parabola  which  is  an  involute  to  OM. 
In  like  manner  any  curve  can  be  described  by 
means  of  a  pattern  of  its  involute.  The  cycloid 
and  ellipse  are  drawn  with  special  facility  by  this 

method.     Thus,  for  the  ellipse,  take  a  thin  rectan-      DOE 
gular   board  ABED,  and  upon  it  fasten   two  pat-  FIG.  162. 

terns  ACOD,  and  BC'OE,  the  edges  CO  and  C'O  being  the  evolute. 
Then  fastening  at  O,  one  end  of  a  string  whose  length  is  AGO,  the  free 
end  will  describe  the  semi-ellipse  as  it  is  moved  from  A  to  B.  Upon  this 
principle  attempts  have  been  made  to  make  a  pendulum  vibrate  in  the  arc 
of  a  cycloid. 

229.  COR. — Every  curve  has  one  and  only  one  evolute  ;  but  every 
evolute  has  an  infinite  number  of  involutes,  since  every  point  in  the  string 
describes  an  involute  as  the  string  unwraps  from  the  evolute. 


SECTION  VIII. 
Envelopes  to  Plane  Curves, 

230.  DEF. — An  Envelope  is  the  locus  of  the  intersection  of 
consecutive  lines,  or  curves,  represented  by  a  given  equation,  when 
one  or  more  of  its  parameters  are  made  variable. 

ILL.— Let  (x  —  m)2  -|-  y~  —  r2  =  0  be  the  equation  of  the  locus  whose  envelope 


ENVELOPES  TO  PLANE  CURVES. 


153 


ct'&'c'd? 

FIG.  163. 


is  required.  Let  m  be  the  (variable) 
parameter.  Let  r  =  Bl,  so  that  Baa' 
shall  be  one  position  of  the  given  locus, 
which  in  this  case  is  a  circle.  Now  sup 
pose  m  to  take  an  infinitesimal  incre 
ment  dm,  putting  the  centre  at  2,  and 
giving  {x  —  (m  -f  dm) }  2  -j-  y*  —  r2  =  0 
as  the  equation  of  the  consecutive  locus. 
The  intersections  of  these  loci,  as  a,  a',  are  points  in  the  envelope.  Again,  let  m 
take  another  infinitesimal  increment,  as  2  3,  then  b  and  &'  are  points  in  the  envel 
ope.  In  like  manner  the  intersections  of  3  and  4,  4  and  5,  etc.,  etc.,  give  points 
in  the  envelope.  The  envelope  in  this  case  is  evidently  the  two  parallel  right 
lines  MN,  M'N'. 

Were  we  to  make  r  vary  at  the  same  time  as  m,  the  form  of  the  envelope  would 
be  changed,  and  would  depend  upon  the  relative  rates  of  change  of  r  and  m.  Of 
Bourse,  the  student  will  understand  that  the  points  of  intersection  a,  6,  c,  d,  etc., 
are  only  in  the  envelope  when  1  2,  2  3,  3  4,  etc.,  are  infinitesimal;  in  other  words, 
the  envelope  is  the  limit  toward  which  these  consecutive  intersections  approach  as 
the  increments  2  3,  3  4,  etc. ,  diminish. 


231,  I*rob. —  To  find  the  equation  of  the  envelope  of  a  given  locus. 

SOLUTION.  — Let  F(x,  y,  m)  =  0  be  the  equation  of  the  given  locus.  The  consec 
utive  locus  will  be  F^x,  y,  m  -\-  dm)  =  0,  or  F(x,  y,  m)  -\~  dmF\x,  y,  m)  =  0.  If  we 
now  combine  the  equation  of  the  locus  with  this  equation  of  its  consecutive,  elim 
inating  m,  we  shall  determine  the  locus  of  the  intersection,  i.  e.,  the  envelope. 
But  since  F(x,  y,  m)  =  0,  the  equation  of  the  consecutive  can  always  be  reduced 
to  dmF(x,  y,  m)  =  0.  Hence  in  practice  we  simply  combine  the  equation  of  the 
locus  with  its  first  differential  equation  eliminating  the  parameter,  thus  obtaining 
the  envelope.  Q.  E.  B.  - 

Son. — It  is  of  course  possible- that  the  consecutive  loci  may  not  intersect ; 
as,  for  example,  xz  -f-  y2  =  ?*2,  when  r  is  made  variable. 

Ex.  1.  Find  the  envelope  of  ?/2  =  m(x  —  m}. 

SOLUTION. — Differentiating  with  reference  to  m,  we  have  0  =  dm(x  — m)  —  mdm, 
or  0  =  xdm  —  2maVn.  Whence  m  =  £.r.  Combining  this  with  y2  =  m(x  —  m),  so 
as  to  eliminate  m,  i.  e. ,  substituting  \x  for  m,  we  have  y  =  ±  $x,  as  the  equation 
of  the  envelope. 

.  ILL. — The  geometrical  significance  of  this  operation 
will  be  readily  seen  by  constructing  a  few  parabolas  on 
the  same  axis,  giving  to  m  slightly  differing  values.  The 
consecutive  intersections,  a,  6,  c,  a',  &',  c',  etc.,  will  ev 
idently  approach  the  two  straight  lines  AM,  AM' as 
the  difference  between  the  Consecutive  values  of  m  is 
made  less  ;  therefore  these  lines  are  the  envelope  of  the 
parabola  y2  =  m(x  —  m),  or  the  series  of  consecutive  pa 
rabolas  of  which  four  are  represented  in  the  figure. 


Mr 


FIG.  164. 


154 


PROPERTIES   OF  PLANE   LOCI. 


Ex.  2.  Find  the  envelope  of  y  =  ax  -\ ,  a  being  the  parameter. 

Construct  a  figure  illustrating  the  result. 

The  envelope  is  y*  =  4m.r. 

Ex.  3.  A  line  of  fixed  length  slides  between  two  fixed  lines  at  right 
angles  to  each  other  ;  required  the  envelope. 

SOLUTION. — Let  the  axes  AX  and  A  V  be  the  fixed 
lines  at  right  angles  to  each  other,  between  which  the 
line  M  N  of  fixed  length,  as  c,  slides.  Let  A  M  =  b, 


and  A  N  =  a,  whence  tan  M  N  X  = ,  and  the 

a 


M 


N 
FIG.  165. 


equation  of  M  N  is  y  = x  -\-  6,  or  -  -\ —  =  1.    (1). 

We  have  also  a*  -f-  62  =  c2  (2).     The  most  direct  method 

(not  the  most  expeditious)  would  now  be,  to  find  the 

value  of  a  or  b,  from  (2),  and  substitute  it  in  (1),  which 

would  then  have  but  a  single  parameter  and  its  envelope  could  be  found  as  before. 

But  the  following  method  is  less  tedious  :    Differentiating  (1)  and  (2)  we  have 

^  -j-  ^  =  0  (3)  ;    and  ada  -f  bdb  =  0  (4). 
Whence  by  eliminating  da  and  db  between  (3) 

and  (4),  we  have  -  =  -3f ;  which  substituted  in 
a         b3 

(1)   gives  after  reduction    b3  =  c'2y.       Similarly 
a3  =  c2«.     Substituting  these  values  in  (1),  there 


results  y    -f~  x 


c  ,  the  equation  sought. 


ILL. — This  locus  is  readily  sketched  by  drawing 
M  N  in  slightly  changed  positions,  and  noting  the 
intersections  of  consecutive  lines,  as  in  Fig.  166. 

SCH. — This  locus  is  a  variety  of  Hypocydoid, 
a  kind  of  curve  generated  by  a  point  in  the  cir 
cumference  of  a  circle  rolling  on  the  concave  arc  of 
(within)  a  fixed  circle.  In  this  variety  the  radius 
of  the  fixed  circle  is  4  times  that  of  the  genera 
trix. 


232.  Prop. — The  envelope  to  a  plane 
curve  is  tangent  to  each  of  the  intersecting 
carves  of  the  series. 


FIG.  167. 


DKM.— Let  F(.x,  y,  m)  =  0  (1),  be  the  given  locus.     But  from  dmftx,  y,  m)  =  0 
(2),  we  have   m  =  <p(x,  y)  ;    whence    the    equation  of  the   envelope    becomes 


F{ 


0  (2, ).     If  now  -—  is  the  same  for  both  the  locus  and  its  en- 


ENVELOPES  TO  PLANE  CURVES, 


155 


velope,  it  follows  that  they  have  a  common  tangent,  wherever  they  have  a  common 
point.     From  (1)  we  get  by  differentiating 

dF(x,  y,  m) 


dF(x,  t/,  m)        dF(a,  y.  m)      dy  dy 

-}- •  — -  =  0,  from  wnicn  -- 

dy  dx  dx 


dx 


Differentiating  (2J  we  have 
dF{x,  y,  <p(x,  y}}    .    dFJx,  y,  <p(x,  y)} 


dy  dx' 


dF\x,  y, 


dF(x,  t/,  m) 


,  y) 


d<p(x,  y) 
But  as  by  (2)  dmF{x,  y,  (p(x,  y)}  =0,  and  <p(x,  y}  =  m,  this  becomes 

dF(x,  y,  m) 
dF(x,  y,  m) 


dx 


.  -V.   whence  —  =  — ,  the  same  as  in  the  equation  of  the 

dy  dx  dx  dF(x,  y,  m) 


locus. 

Ex.  4.  Find  the  locus  to  which  the  hypot-    Y 
enuse  of   a  right   angled    triangle  of   con 
stant  area  is  always  tangent. 

SOLUTION.— Let  the   constant  area  ABC 


a, 

Then    AB    =  --, 
m 


the    parameter    AC    =  m. 

tan  BCX  = ,  and  the  equation  of   BC  is 

w2 

y  = x  H .     The  equation  sought  is  xy  —  -, 

the  equation  of  an  hyperbola. 


C 

FIG.  168. 


Ex.  5.  What  is  the  envelope  of  an  ellipse  which  retains  its  axes  in 
the  same  right  lines,  but  varies  in  eccentricity  so  that  AB  =  a  con 
stant,  m? 


SUG'S.  —  Since  AS  =  m,  the  equation  of  the  ellipse  is  A*y*  -f-  msx2  =  Aym^  ;  in 
which  A  is  the  parameter.  The  equation  of  the  envelope  is  xy  =  £ra,  an  equilat 
eral  hyperbola  referred  to  its  asymptotes. 

As  will  appear  hereafter,  the  area  of  an  ellipse  is  itAB.  Hence  the  area  of  the 
above  locus  is  constant. 

Ex.  6.  From  every  point  in  the  circumference  of  a  circle,  pairs  of 
tangents  are  drawn  to  another  circle.  Find  the  locus  to  which  the 
chord  connecting  corresponding  points  of  tangency  is  constantly 
tangent. 

SOLUTION.—  Letting  C  be  the  centre  of  the  first  and  A  of  the  second  circle,  it 
is  evident  that  as  P  moves  around  the  circle  P'P"  will  change  its  position.  The 
envelope  of  P'P"  is  required.  Let  CP  =r,  and  AP"  —  r'.  Let  P  be  designat 
ed  as  (m,  n),  P'  as  (m',  n'),  and  P"  as  (m",  n"). 

The  equation  of  the  locus    P'P"  whose  envelope  is  required  is  y  —  n'  = 


156 


PROPERTIES   OF  PLANE  LOCI. 


— ; ~(x  —  m')    (1).     But  by  reason  of 

7H     —  ?/i 

the  tangents  PP'  and  PP"  we  have 
nri  -f-  mm  =  r'2  (2)  ;  and  nn"  -f-  mm"  = 
r'2  (3).  Subtracting  (3)  from  (2)  we  have 
n(ri  —  n")  -f-  m(m  —  m")  —  0  ;  whence 


(x  —  m' ),  ny  -f  mat  =  r1*.    Thus  we 


FIG.  169. 

ny  -[-  mx=r'2  (4). 


find  the  equation  of  the  given  locus  P'P"  to  be 

Again,  if  we  let  the  distance  between  the  centres  of  the  circles  AC  be  repre 
sented  by  a,  we  have  the  relation  between  n  and  m  in  the  equation 
n2  +  (m  —  a)2  =  r2     (5). 

The  problem  then  is  to  find  the  envelope  of  (4),  when  the  relation  between  n 
and  m  is  that  given  in  (5).     Differentiating  (4)  and  (5)  considering  m  and  n  as 

nx 
ana  n-; h  m  —  a  =  U.     .  • .  — 

y 

m  —  a  =  0,  my  —  nx  =  ay      (6). 

Finally  eliminating  m  and  n  between  (4)  (5),  and  (6),  and  reducing,  we  have 
yiyi  _|_  (j#  _  a4)a;a  _j_  2or'ia?  =  r'4,  as  the  equation  of  the  envelope. 

Hence  the  envelope  is  a  conic  section.     When  a  =  0  it  is  a  circle  ;  when  a  <  r, 
an  ellipse  ;  when  a  =  r  a  parabola  ;  and  when  a  ^>  r,  an  hyperbola. 


variables,  we  have  y~  4-  x  =  0,  — ^  —  —  -,  and  n—  4-  m  —  a  =  0 
dm  dm  dm  ~ 


IP-rob. — An  infinite  number  of  parallel  right  lines  meet  a 
given  curve  on  the  same  side;  and  where  each  meets  the  curve  a  line  is 
drawn  making  an  angle  with  the  parallel  which  is  bisected  by  the  normal 
at  that  point.  Required  the  envelope  of  the  line. 

SOLUTION.— Let  MN  be  the  given 
curve,  PO  one,  of  the  parallels,  PQ  the 
normal,  and  PG  the  locus  whose  envel 
ope  is  sought.  Our  first  purpose  is  to 
find  the  equation  of  PG.  Let  y'  =  <p;z') 
be  the  equation  of  M  N .  and  r  the  tan 
gent  of  the  constant  angle  PSX.  Since 
P,  whose  co-ordinates  are  #',  y',  is  a  point 
in  PG,  the  equation  has  the  form 
y  —  y'  =  a(x  —  x')  in  which  a  is  the  tan 
gent  of  PDX.  Now  PDX  =  PQX  —  DPQ 

(PSX  PQX)     =     2PQX     -     PSX. 

tan  2PQX  -  tan  PSX 
1  +taniPQXtanPSX'     Again>  aS  PQ  1S  n°rmal 


N 


M 


I 


Fio.  170. 
PQX  -QPS  =  PQX 

Therefore    tan  PDX    — 


fco     '  = 


tan  PQX 


-  j-,  ;  whence  tan  2 PQX 


2  tan  PQX 
1  —  tan*  PQX 


_2 

dy' 


Substituting,  and 


ENVELOPES  TO  PLANE  CUHVES. 


157 


introducing  v  for  tan  PSX,  we  have 


rfx'a  dx' 
——v  —  v  —  2 — 
dy  * d£ 

1        dx'*         dx   ' 

1  -     ~    v 


dx' 


Putting  —7  =  p,  for  convenience,  the  equation  of  PG  becomes 


y  —  y 


:(*  — 


From  this  equation,  its  first  differential  equation,  and  the  equation  of  the  curve 
y'  =  <p(x'\  if  x't  and  y'  be  eliminated,  the  resulting  equation  between  x  and  y 
will  be  the  equation  of  the  envelope  sought.  But  the  difficulties  of  elimination 
are  often  insurmountable.  We  give  two  cases  which  are  readily  solved. 

234.  COR.  1.  —  If  O  P  is  parallel  to  AX,  v  =  0,  and  the  equation 
PG  becomes 


235.  COR.  2.  —  If  O  P  is  perpendicular  to  A  X,  v  =  oo,  and  the  equa 
tion  becomes 


236.  Sen. — It  is  a  well  known  prop 
erty  of  light  that  its  rays  impinging  upon 
a  reflecting  surface  are  thrown  off  so  as 
to  make  the  angle  between  the  reflected 
ray  and  the  normal,  equal  to  that  between 
the  incident  ray  and  the  normal.  In  con 
sequence  of  this  law,  when  the  rays  of  the 
sun,  which  are  practically  parallel,  are  re 
flected  from  a  curved  surface,  the  inter 
sections  of  the  consecutive  reflected  rays 
produce  a  luminous  curve,  called  a  Caustic, 
which  is  an  example  of  the  envelope  dis 
cussed  in  the  problem.  The  annexed 
figure  affords  an  illustration.  Let  NAN' 
be  a  section  of  a  circular  cylindrical  mir 
ror,  made  perpendicular  to  its  axis.  Let 
1  to  11  be  rays  of  light  parallel  to  the 
axis  of  the  mirror  AC.  The  envelope  of 
the  reflected  rays  is  the  caustic  curve  NM. 
MN  shows  the  lower  branch  of  the  caus 
tic,  the  rays  not  being  represented.  This 
curve  may  be  seen  inside  of  a  ring  lying 


Via.  171. 


158  PROPERTIES   OF   PLAXE   LOCI. 

on  a  table  in  the  light.  It  is  familiar  to  the  milkman,  as  "the  cow's  foot 
in  the  milk,"  which  is  the  caustic  formed  upon  the  smooth  surface  of  the 
milk  in  a  bright  tin  pail,  by  reflection  of  the  light  from  the  inside  of  the 
pail. 

Ex.  1.  To  produce  the  equation  of  the  caustic  when  the  incident 
rays  are  parallel  to  the  axis  of  a  parabolic  reflector. 

SOLUTION.—  We  have  y'2  =  4mx',  and  ~-f  =  —,  4ra  being  the  parameter  of  the 

parabola.     Substituting  this  value  of  p,  and  for  x',  ~,  in  the  equation  (234],  we 

have  after  reduction 

yy'z  —  4m?y  -{-  4mY  =  4my'a?      (1). 

2rax  —  2m2 
Differentiating  (1)  with  respect  to  the  parameter  y  ,  gives  y  -       —  -  -  .     bub- 

stituting  this  in  (1),  and  reducing  we  have 

x  =  m  ±  V  —  y'2,  as  the  equation  of  the  caustic.     This  can  only  be 
satisfied  for  y  =  0,  x  =  m  ;  whence  we  see  that  the  caustic  is  a  point,  the  focus. 

Ex.  2.  To  find  the  caustic  to  the  circle  when  the  incident  rays  are 
parallel  to  the  axis  of  x. 

SOLUTION.—  Equation  of  circle  y'2  -f  of*  =  r2.      .-.—=—  y-.    Equation  of 
reflected  ray  (234)  y—y'  = 


—  *'),  becomes  y  —  y'=-    , 


<**-*'  =  -(y'  -  y}>  or'  ^-x  =  y- 

Differentiating  (1)  with  respect  to  £',  we  have 


'  -    i 

-^—  «-?*}• 


=          *:y  _  y)  ;    vrhich  divided  by  -  gives 

2/'2  y'aV       a;'2      /  %- 

,.'2  r2  |    i 

—  ==  —  (w'  —  y).     From  which  we  find  y'  =  r  ?/'  . 
S/'*       2/'3 

To  find  a',  substitute  in  (1)  for  y'  —  y  its  value  '-^-,  and  we  have 

y'2-*'2  ^  =  y'2  -  tt'2x,    But  x.,  =  r2  _  ,2  wbence  §r  _  ^  = 

2y'x'        r2  2ra 

~r/a  -VV,  or  x'  =  2_T'^  ,g.     Putting  r3y    for  y',  this  becomes  x'  =  -^  —  •—. 


BECTIFICATION  OF  PLANE  CURVES.  159 

Finally,  squaring  these  values  of  y',  and  x  and  substituting  in  y'2  -\- x'*  =  r2,  we 

4.  .a             4r3r2 
have  r*y3  -j ^ : — —  =  r2,  which  is  the  equation  of  the  caustic  sought. 


[NOTE.— At  this  stage  of  the  course  the  student  will  need  to  acquaint  himself  with  the  elements 
of  the  Integral  Calculus,  as  given  in  Chap.  lit   of  the  second  part  of  this  volume.] 


SECTION  IX. 

Kectification  of  Plane  Curves, 

(a)    BY  MEANS  OF  RECTANGULAR  CO-ORDINATES. 

DBF. — To  Rectify  a  Curve  is  to  find  its  length,  The 
term  arises  from  the  conception  that  a  right  line  is  to  be  found  which 
has  the  same  length  as  the  curve. 

238.  Prop. — The  formula  for  the  rectification  of  plane  curves  is 

dz  =  \/dx2  +  dy2 ; 

in  which  z  represents  the  length  of  the  curve,  and  x  and  y  the  general  co 
ordinates. 

DEM. — Let  M  N  be  any  plane  curve,  AD 
=  x,  and   P  D  =  y  be  any  co-ordinates,  and 
let  D  D'  represent  dx  ;  then  will  P'  E  repre 
sent  dy,  and  PP',  dz  ;  ?,.  e.,  dx,  dy,  and  dz  will 
represent   contemporaneous   infinitesimal  in 
crements   of    the   co-ordinates    and   the   arc.    X^T 
From  the  right    angled  triangle    PEP'   we     R 
have  at  once     dz  =  Vdx-  -j-  dy2.     Q.  E.  D.  FIG.  172. 

239.  SCH. — To   apply  this    formula  to  any  particular   curve,  we  have 
simply  to  find  dx  or  dy  from  the  equation  of  the  proposed  curve,  substitute 
it  in  the  formula,  and  then  integrate  between  proper  limits. 

Ex.  1.  Rectify  the  semi-cubical  parabola  whose  equation  is  y*=zax3. 
(See  evolute  of  common  parabola.) 

DEM. — Differentiating  y'2  =  ox3,  we  have  dy  =  — —  dx,  whence  dy-  =  - — ~dx*  = 


. 
— — -dx1*  —  %axdx*.     Substituting  this  value  in  the  formula  for  rectification,  it 

becomes  dz  —  (dx2  -f   %axdxi)     =(!-)-  fax)2efce.      Integrating  we  have  z  = 


160 


PROPERTIES    OF   PLANE   LOCI. 


—  (1  -f  fax)2  +  C.     To  determine  C  we  may  reckon  the 
fjo 


length  of  the  curve  from  the  origin  A,  whence  for  x  =  0, 

8  8 

z  =  0,  and  we  have  0  —  --  \-  C,  or  C==  --  . 

',>7/»      '  O7/. 


27a 
I 


The  cor- 

To 


reded  integral  is  therefore  z  =  ^=-[(1  +  $a.x)'2    —  1]. 

illustrate  this  result  consider  A  M  the  curve  whose  length 
is  to  be  found,  or,  which  is  to  be  rectified.  As  this  curve 
is  infinite  in  extent,  we  can  only  inquire  for  the  length  of 
some  specified  portion  of  it.  Let  it  be  required  to  find 
the  length  of  the  curve  between  the  origin  and  the  point 
P  whose  abscissa  we  will  call  6.  Substituting  this  value 

ft  3. 

of  x,  we  have  arc  AP  —  z  =  ^=-[(1  -f  f«6)2  —  1].     Were 


D' 


FIG.  173. 


it  required  to  find  the  length  of  some  other  arc,  as   PP',  we  should  integrate 
between  the  limits  x  =  AD,  and  x  =  AD'.     Thus,  let  AD  =  6  and  AD'  =  c. 

n  ^ 

Resuming  the  indefinite  or  general  integral  z  =  t-r=-(l  -j-  \axf  -f  C,   substituting 

Bf  tt 

successively  x  =  b,  and  x  —  c,  and  subtracting  the  former  result  from  the  latter, 
we  get  for  the  length  of  the  arc  P  P',  the  definite  integral 


Ex.  2.  Rectify  the  common  parabola. 

SOLUTION.  —  From  T/?  =  2px  we  have  dxz  =       '*•  ;    whence  dz  =  -(p*  -f-  y2)  dy. 
To  integrate  this  apply  formula  <g  of  reduction,  and  we  have 


y\/p- 


Estimating  the  arc  from  the  vertex,  which  is  the  origin,  we  have  C=  —  4logp  ;  and 


wv/p'2  4-  y2       p      rv  -f-  v/p2  -f-  v>2~| 

the  corrected  integral  is  z  =  y-  —  '  —  1-i.  _|_  ^-  W    LJLJ-JL.__LJL  I 

2p  r  2     b  L  p  J 

SCH.—  Instead  of   integrating  as  above,  we  may  expand  (p*  -f  £*)*  by 
Maclaurin's  or  the  Binomial  theorem,  and  then  integrate  each  term  separate- 

.         1    1    7/3       1     1    1    1    r      1    1    3    1    1    1    y7 

ly,  obtaining  «==(«/  4-  -  •  -  •—  --  .-._._.__L.-.-.-.._._._.^-  —  etc.  ) 
U^2   3   ;r2      2225  jo^2    22237^ 


Ex.  3.  Rectify  the  circle. 

SOLUTION.  -  From  x2  -}-  y2  =  r2  we  have  dy*  =.  —^dxy,  hence  dz  =  (X     ^      ) 


RECTIFICATION   OF  PLANE   CURVES. 


161 


dz  =  --  '  -  -,  and  z  =  rsin—  '-  -f-  C.      But   this  is  only  a  restatement  of  the 

(i*-*)1 

problem  and  is  of  no  use  in  the  solution.     We  shall  have  to  integrate  in  some 
other  way.     We  may  write 

dz  =  r(r2  —  SB*)~*(£C  ~ 


for  convenience.     Expanding  by  Maclaurin's  or  the  Binomial  the 


, 
_1        etc 


putting  -  = 

orem,  and  integrating  each  term  separately,  we  have 

a?,  3  3*,5       ,    1  •  S.S 

+  a_L.  +  _i_   +  __ 

Bestoring  x  this  becomes 

x    .        x*  3x°  1-3 


" 


23  2  •  4  .  6  .  7r 

To  determine  (7,  reckon  the  arc  from  the  axis  of  ordi- 
nates  (B),  whence  for  x  =  0  z  =  0,  and  therefore  C—  0. 
Then  making  x  =  r  we  have  the  length  of  the  quadrant 

1  1-31.3-5 

_  +  f^  +  ^7T7T 


BX,  z  = 


etc. 


.  Bepresenting  the  sum  of  the  series  in  the  parenthesis 
by  i^r,  we  have  z  =  a  quadrant  =  ^TTC,  and  the  whole  cir 
cumference  =  2;rr.  Letting  D  be  the  diameter,  this  be 
comes  DTT,  whence  it  appears  that  n,  the  sum  of  the  series 
above,  is  the  ratio  of  the  circumference  of  a  circle  to  its 
diameter.  By  extending  the  terms  in  this  series  sufficiently,  reducing  each  to  a 
decimal  fraction  and  adding,  we  find  Tt  =  3.14159'26-f-.  For  practical  purposes  TC 
is  usually  taken  as  3.1416,  and  for  still  ruder  approximations  as  3^. 


G"         ' 


.  COR.  1.  —  The  circumferences  of  circles  are  to  each  other  as  their 
radii,  or  as  their  diameters. 

241.  SCH.  —  The  quantity  Tt  has  not  only  fundamental  importance  in  ge 
ometry,  but  has  great  historic  interest.  Upon  it  depend  both  the  method 
of  obtaining  the  circumference  of  a  circle,  of  a  given  radius,  and  the  area 
of  a  circle,  as  well  as  many  other  problems.  The  ancients  sought  with 
much  diligence  to  discover  its  value.  Archimedes  (287  B.  C.  )  found  it  to 
be  between  3J7?  and  3}.  Metius  (1640)  gave  a  nearer  approximation  in  the 
fraction  $ff.  In  1853  Mr.  Kutherford  presented  to  the  Boyal  Society  of 
London  a  computation  by  Mr.  W.  Shanks  of  Houghton-le-  Spring,  extend 
ing  the  decimal  to  530  places.  The  following  is  its  value  to  50  places  : 

3.141  592  653  589  793  238  462  643  383  279  502  884  197  169  399  375  10. 


4.  Rectify  the  cycloid. 


SOLUTION.  —  We  have  de2  = 


^    ^ 


,  whence 


*s  j 

fdz  =  (2r)-/(2r  —  y)~kdy  =  —  2(2r)^(2r  —  y)*  -f  C. 
Beckoning  the  arc  from  the  origin   C  =.  4r,  and  the  corrected  integral  is  z 


162 


PROPERTIES   OF   PLANE   LOCI. 


—  2(2r)  (2r  —  ?/)    -f  4r.     Making  y  =  2r,  z  =  £  the  cycloidal  arc  =  4r,  whence 
the  entire  arc  of  the  cycloid  is  seen  to  be  4  times  the  diameter  of  the  generatrix. 

242.  COR.  2. — Any  arc  of  a  cycloid,  estimated  from  the  vertex,  is 
equal  to  twice  the  corresponding  chord  of  the  generatrix. 

DEM. — Resuming  the  indefinite  inte 
gral  z  =.--  2(2r)  *(2r  —  ?/)  +  C,  if  we 
estimate  the  arc  from  B ,  where  y  =  2r, 
we  have  C  =  0  ;  and  the  corrected  in 


tegral  is  z  =  —  2(2r)   (2r  —  y)  . 


This 


is   the  length   of   any   arc   estimated 

from    B,  as  BP,  PD  being  y.     But  FIG.  175. 

in   the   right  angled    triangle    BEC,    BE  =  \/BC   X    BQ   =  vx2r(2r y). 

.  • .  arc  B  P  —  2  times  chord  B  E.     Q.  E.  D. 

Sen.— Both  the  fact  that  the  length  of  the  cycloid  equals  4  times  the 
diameter  of  the  generatrix,  and  that  any  arc  estimated  from  the  vertex 
equals  2  times  the  corresponding  chord  of  the  generatrix,  are  readily  ob 
served  from  the  manner  in  which  the  curve  is  described  from  its  evolute. 
Thus  the  radius  of  curvature  at  the  vertex  is  2  times  the  diameter  of  the 
generatrix.  But  this  is  the  arc  of  the  evolute.  So  also  as  the  string  un 
winds  from  the  evolute,  the  radius  of  curvature  is  seen  to  be  the  arc  of  the 
cycloid  and  equal  to  twice  the  corresponding  chord  of  the  generatrix. 
(See  Fig.  159.) 

Ex.  5.  Rectify  the  hypocycloid,  whose  equation  is  a?  +  7/71  =  a?. 
(See  Fig.  167.)  The  length  of  the  entire  curve  is  60. 

Ex.  7.  Rectify  the  ellipse. 
SOLUTION. — For  this  purpose  the  equation 

y*  -f    (1    _   e2)a.2   _   ^2(1    _   g2) 

is  most  convenient.      From  this  we  have   dy*  =  — •{! e*)*dx**  whence 


—     \dx*  +  -n  — 


-f  x2(l  —  e*)2 


—  x*)  -f  (1  — 


(1  — 


—  a;2) 


=  dx J^g  =        AdX  -     \l-~. 
But  this  form  can  only  be  integrated  approximately.     Developing  (l  —  -  ~  j  ,  we 


have    dz  = 


A  C  dx    -~  C  x'dx  _ 


RECTIFICATION   OF   PLANE   CURVES. 


163 


—  etc.     Now  integrating  each  of  these  terms  separately  we  have  z  =  A  sin—1—  — 

A 


3e6        (  SA^ZAi/A1  .       x       x  /-  -  \       sr3    ,-  -  ~|       x* 
-  arreSt  TrU-(28m-'a  -f/A'-*)-  jV^-^-g- 


—  etc.,  -f-  C. 

If  we  estimate  the  arc  of  the  ellipse  from  the  extremity  of  the  conjugate  axis, 
we  have  for  x  =  0,  z  =  0  ;  whence  substituting,  we  find  (7=0. 

Again,  making  x  =  A,  and  observing  that  sin-1!  —  -,  we  have 


e*/AjtA 


"~~       ~ 


Uniting  and  multiplying  by  4,  we  have  for  the  entire  circumference  of  the  ellipse, 

./.,  e2  3e4  3  .  3  .  5e« 

4.,  =  2^(1  -  272  -  -a.a.4.4  ~  2.2.A.4.6.6  -  • 

This  series  converges  more  or  less  rapidly  as  the  eccentricity  is  greater  or 
but  is  always  converging. 


(b)    RECTIFICATION  BY  MEANS  OF  POLAR  CO-ORDINATES. 
243.  JPTOp. — The  formula  for  rectifying  polar  curves  is 

dz  =  (r*d02  +  dr2)"5" 

DEM.— Let  A  be  the  pole  of  the  curve   MN,  AP  any  radius  ^ 

vector,  and  AP'  the  consecutive  position  of  the  radius  vector,  so  D^ 
that  PAP' =  d0,  0  being  the  variable  angle.  Let  z  represent  any 
arc  of  the  curve,  and  r  the  radius  vector,  and  with  A  as  a  centre  and 
radius  AP,  draw  PD.  Then  PP'  =  dz,  and  P'D  =  dr,  are  infin 
itesimals  of  z  and  r  respectively,  and  contemporaneous  with  dQ. 
Now  from  the  right  angled  triangle  P'DP,  right  angled  at  D,  we 

.   / *-         I=T 

have  P'P  =    \PD    -|-  P'D  .     But  dQ  being  the  arc  measuring 

PAP'  at  a  unit's  distance  from  A,  PD  =  rdQ  ;  whence,  substitut 
ing  dz  for  PP',  dr  for  P'D,  and  rdQ  for  PD,  we  have  dz  =  (r2d02  -f-  dr2)  . 
Q.  E.  D. 

Ex.  1.  Rectify  the  logarithmic  spiral,  log  r  =  0. 


FIG.  176. 


SOLUTION.     d02  = 


.  • .  dz  =  ( M*  -f  1)  dr,  and 


z  =  (JJf«  +  1)  r  -f  C.  If  the  arc  be  reckoned  from  the 
pole,  so  that  z  —  0  when  r  =  0,  the  constant  C=  0,  and  the 

corrected  integral  is  z  =  (M*  -f  1)  r.  If  we  take  the  Na 
pierian  logarithm  of  r,  we  have  z  =2  r.  Now  when  6  =  0 

r  =  1,  and  z  =  2  .  But  by  tracing  this  curve  we  see  that 
if  A  B  Fig.  177,  represent  the  value  of  r(  =  1)  when 
6  =  0,  there  are  an  infinite  mimber  of  spires  between  this 


FIG.  177. 


164  PROPERTIES  OF  PLANE  LOCI. 

and  the  pole  (see  119\  and  we  have  the  singular  result  that  their  entire  length  is 
\/2.  When  0  =  2tf  =  6.2831853-f  ,  we  have  log  r  =  6.2831853-f  ,  and  in  the  com 

mon  system  7-  =  1919487.61-(-   and   z  =  (M*  -f  1)     X  1919487.61-f,   in  which 

(M*  -f  1)*  =  [(.4243-f  )2  If  I]4  =  1.086-f.  Here  we  have  another  singular  result, 
viz.,  that  the  whole  of  the  infinite  number  of  spires  within  the  value  r  =  1  (A  B), 
and  the  spire  generated  by  the  revolution  from  0  =  0  to  0  =  2it,  are  together  only 
a  trifle  longer  than  the  radius  vector  after  this  revolution. 

f\ 

Ex.  2.  Rectify  the  spiral  of  Archimedes,  r  —  —  -. 

&7t 

Rzsuli.  —  For   convenience  put  —  =  a,   writing  the  equation  r  =  aO.     Then 

Ait 

j  dz  =  -  j  (r2  -f  #2)  cZr,  and  the  process  of  integration  is  identical  with  that  used 
in  rectifying  the  common  parabola  (Ex.  2,  239). 


log 

b 


2a  2 

Ex.  3.  Bectify  the  cardioid,  r  =  a(l  -f  cos  0). 

SOLUTION.  —  This,  as  the  name  implies,  is  a  heart-shaped  curve.     The  student 
should  first  construct  it.     dr2  =  a'2  sin2  0d02,   and  r2  =  a2  -f  2a2  cos  0  -f  a-  cos2  0  ; 

whence  dz  =  (a2  -f  2ascos0  -f  a2cos20  -f  «2sin20)  dQ  —  a(2  -f  2cosG)  dO  = 
2acos£9cW,  since  1-f  cosQ  =  2  cos2  £9.  (See  Trigonometry  57,  P.)  .•.  z  = 
4a  sin  $Q  +  ^  Estimating  the  arc  from  0=0,  we  have  z  —  0,  whence  (7  =  0; 
and  the  corrected  integral  is  z  =  4a  sin  £0.  Making  0  =  180°  we  have  z  ==  4o. 
This  being  £  the  circumference,  the  entire  length  is  8a. 


SECTION  X. 

Quadrature  of  Plane  Surfaces, 

(a)  BY  EECTANGULAK    CO-OKDINATES. 

24:4:.  DEF. — The  Quadrature  of  a  surface  is  finding  its  area. 
The  term  quadrature  comes  from  the  conception  that  we  find  an 
equivalent  square.  Thus  the  quadrature  of  the  circle  consists  in 
finding  a  square  of  the  same  area. 

24&.  JProp. — The  formula  for  the  quadrature  of  plane  surfaces  is 
dA  =  ydx,  or  xdy. 

DEM. — Recurring  to  Fig.  172,  it  is  proposed  to  find  the  area  of  the  surface  lying 
between  MN  and  AX.  Calling  this  area  A,  the  trapezoid  PP'D'D  included 
by  two  consecutive  ordinates  is  dA,  a  differential  element  of  the  area  contempora- 


QUADRATURE   OF   PLANE   SURFACES.  165 

neous  with    dx,   and  dy,    as  heretofore    considered.       But  area    PP'D'D    = 

—  X  D  D',  or  dA  =  tT^~ —  ^x  —  ydx  H~  k^y^-     Since  the  last 

term  is  a  differential  of  the  second  order  with  reference  to  the  others,  it  must  be 
dropped,  and  there  results  dA  =  ydx.  In  like  manner  an  element  of  the  area 
lying  between  a  curve  and  the  axis  of  ordinates  may  be  shown  to  be  dA  =  xdy. 

Q.  E.  D. 


Ex.  1.  Find  the  area  of  the  common  parabola. 


SOLUTION. — From    y2  =  2px,    we   have   dx  =   -ydy.    Y 


Whence   dA  —  -y"dy,   and  integrating  A  —  f-  -f-  (7. 

Beckoning  the  area  from  the  origin  A,  for  y  =  Q,  A  =  Q, 

V3 

whence  (7=0,  and  the  corrected  integral  is  A  =  ,—  = 

3p 

y*__y  _  2pxy  =  ^ 

3p  3p 


APD  is  IADPE. 

Consequently    PAP'  =  lEPP'E'.      The   latter  rect 


FIG.  178. 


angle  is  sometimes  called  the  circumscribed  rectangle,  and  the  area  of  the  parabola 
is  said  to  be  •§  of  the  circumscribed  rectangle. 

To  find  the  area  of  any  specified  portion,  as  that  between  x  =  A  D  =  a,  and 
x  =  AD"  =  ?>,  or  of  the  surface    PP"  D"D,  we  resume  the  indefinite  integral 
i 

-f-  C  \  substituting  a  and  6  successively  for  x,  and  subtract 
ing  the  former  result  from  the  latter,  i.  e. ,  integrating  between  the  limits  x  =  a, 
X  =  &,  we  have 

\f  /Q\*«2        33  1        t  ^ 


A  =  ~-  4-  C=  — ^ 
Ap  op 


Ex.  2.  Find  the  area  of  y  =  x  — •  x3. 

SOLUTION. — Substituting  the  value  of  y  in  the 
general  formula,  it  becomes  dA  =  xdx  —  x3dx 
.• .  A  =  £.«2  —  Isc4  -(-  C.  Sketching  the  curve, 
we  observe  that  it  will  be  natural  to  inquire  for 
the  area  AmB,  or  Am'B'.  Thus  reckoning 
the  area  from  the  origin,  a;  =  0,  gives A=  0,  and 
consequently  (7=0.  The  corrected  integral  is 

A  =  \&  —  $x4.     Hence  area  AmB,  or  Am'B'  -r,       1Pro 

jj  IG.  1 1  y. 

==  4,     To  find  the  area  of  any  other  portion  as 

BCD,  we  integrate  between  the  limits  x  =  A  B  =  1,  and  x  =  A  D  = 

gives  A  =  2  —  4  —  4=  —  2.J.     Hence  area  BCD  =  2^. 


2.     This 


Ex.  3.  Find  the  area  of  y  =  .r*  —  bzx 

Area  betiveen  x  =  0  and  x  =  b  is 


166  PROPERTIES   OF  PLANE   LOCI. 

Ex.  4.  Find  the  area  of  y  =  x*  +  a#2,  constructing  the  curve,  and 
observing  the  natural  limits  of  integration. 

Result,  Between  x  —  0,  and  x  =  a,  A  =  T7^a4  ;  and  between  x  —  0 

and  x  =  —  a,      /  dA  =  TVz4. 

«/  —  a 

.  Ex.  5.  Construct  and  find  the  area  of  a2t/2  =  #2(a2  —  #2). 

Area  =  ^a2. 

Ex.  6.  Required  the  quadrature  of  ay*  =  a3  between  the  limits 
y  =  6,  and  y  —  c.  Area  =  2a3 

Ex.  7.  Required  the  quadrature  of  the  circle. 

SOLUTION.  —  From  x2-f  y2  =  r2,  we  have  y=(r2  —  x2)  ,  hence  dA  =  (r2  —  a2)  da. 
Applying  formula  <JJ  of  reduction  and  integrating,  we  find  A  —  £x(r2  —  x2)2  -f- 
ir2sin—1  —  (-  (7.  Estimating  the  arc  from  the  axis  of  ordinates,  so  that  for  x  =  0, 

A  =  0,  C  is  also  0,  and  the  corrected  integral  is  A  =  £x(r2  —  x2)'2  -f-  ^r2sin-1-. 

Making  x  =  r,  we  have  area  o/  4  o/  Me  circ/e  =  4  r2?r  ;  whence  the  entire  area 
of  a  circle  whose  radius  is  r,  is  nrr2.  Now  it  has  been  determined  with  sufficient 
accuracy  in  Ex.  3,  ART.  239. 

Without  assuming  the  value  of  it,  we  may  expand  (r2  —  a;2)*  and  integrate 
approximately.  Thus  dA  =  (r2  —  a2)  *  da;  ==  r(  1  —  ^  V  dx,  and  putting  -  =  x  ,  ,  we 


whence  A  =  r2[/dxL  — 

yJs/a;,  '"dx!  —  Tt&T/*!  I2^i  —  etc.] 

=  rtfa?    -  ^i-3  _  £i!  -  ^L_  _  5?li  _  Z^ii1  _  21x>  13  _        1    ,    /, 
L    l         6          40          112        1152        2816        13312  J  "*"    ' 

in  which  C  is  0  when  the  arc  is  estimated  from  the  axis  of  y.     Now,  making  sc=rt 
makers  x,  =  1  ;  whence  the  area  of  4  the  circle  is 

A  =  r*(l  -  i  -  VIF  -  T}?  -  T-A-?  -  FJ¥  -  T-3Vr]r  -  etc.). 
The  sum  of  this  series,  thus  extended  is  1  —  .208999  =  .791001.     Hence 

Area  of  circle  =  4  X  r2  X  .791001  =  3.164005r2,  approximately.  This  number 
3.164005,  if  more  accurately  computed  is  found  to  be  exactly  the  same  as  it, 
Rv.  3,  ART.  239.  A  nearer  approximation  can  readily  be  made  by  extending  the 
series. 

240.  COR.  1.  —  The  areas  of  circles  are  to  each  other  as  the  squares  of 
their  radii. 

DEM.—  Let  r,  and  r,  be  the  radii  of  two  circles.  Their  areas  are  tfr2  and  itrj. 
But  itr*  :  Tir^  :  :  r2  :  r^. 

.  COR.  2.  —  The  area  of  a  circle  whose  radius  is  1  is  TT. 


QUADRATURE   OF  PLANE   SURFACES.  167 

?.  COR.  3. —  The  area  of  a  circle  is  equal  to  its  circumference  into 
%  its  radius. 
DEM.     r27T  =  2rnr  X  ^r,  and  2r;r  is  the  circumference  (239,  Ex.  3). 

240.  3* rob. — To  find  the  area  of  any  segment  of  a  circle. 

SOLUTION. — Let  the  distances  from  the  centre  to  the  bases  of  the  segment  be  6 
and  a,  so  that  a  —  &  =  the  height  of  the  segment.  We  have  but  to  resume  the 
indefinite  integral  and  take  its  value  between  these  limits.  Thus 

j  A    -I-/y/'t*2  n$\          I      JL^*>  i-.^-*-* — i  1  Tv/A*9.  7*9\  l./v.o  (-,'•*•*  —  i 


y 

Jx  =i 


r  r 

If  the  segment  is  reckoned  from  the  diameter,  6=0,  and 


C 
I 

* 


=  L  a 

dA  =  £«(r2  —  a2)2  -|-  £r2sin-J-.     The  significance  of  this  is  readily  seen  by 


4 
inspecting  Fig.  174.     AD  =  a,  and  £a(r2  —  a8)    =  area  of  APD.     Arc  BP 


r  sin—'—  ;  whence  -i-r2  sin"1—  =  area  sector  A  B  P. 
r  r 

If  the  segment  has  but  one  base,  a  =  r,  and  we  have 

r  i.  7> 

A  =  iTtr*  —  £b(r*  —  &*)*  —  £r2  sin-)-.     From  Fig.  174,  we  see  that  Itfr2  =  4 

r 

the  area  of  the  circle,  |r2sin—  <-  =  area  BPA,  and  ^b(r~  —  62)2  ==  area  APD. 
In  each  case  we  have  £  the  segment. 

Ex.  8.   Find  the  area  of  the  ellipse. 

7?  rA         *-  rA         A- 

SUG'S.—  The    area  =  —  /    (A2  —  x2)2dx.    But    /    (A*  —  x2)  cZx  is  4  the  area  of 

AJ*  Jo 

a  circle  whose  radius  is  A,  which  is  TtA2.     .  •  .  Area  of  ellipse  =  itAB. 

25O.  COR.  —  The  area  of  an  ellipse  is  to  the  area  of  the  circumscribed 
circle  as  the  conjugate  axis  is  to  the  transverse  axis.  The  area  of  an  ellipse 
in  to  the  area  of  the  inscribed  circle  as  the  transverse  axis  is  to  the  conjugate 
axis.  The  area  of  an  ellipse  is  a  mean  proportional  between  the  inscribed 
and  circumscribed  circles. 

Ex.  9.  Find  the  area  of  the  cycloid. 

SUG'S.—  We  have  dx  =  -  ^  —  r  ;    whence  dA  =  _JSS  —  _   = 

(2ry  —  2/2)*  (2ry  —  yrf 

y  (2r  —  y)  *dy.     Integrating  by  applying  formula  ^  twice,  we  obtain 


=    /*V(2r  — 


=  f  r2  Ters-'2        Y 


=  f  r^Tt,  .since  vers-]2  =  it.  .  •  .  The 
entire  area  is  3^r2,  or  three  times  the 
generalmy  circle. 

A   somewhat    indirect    but    simple        A  A'  EX 

method  of  quadrature  is  as  follows  : 

1st.  Find  the  area  of  A  PCS,  the 


1G8  PROPERTIES  OF  PLANE  LOCI. 

element  of  which  DPP'D'  =  dz(2r  —  y)  =  (2r  —  y)  -  ^--  —  z  =  (2ry  —  ys)*dy. 


Now  considering  the  circle  A'pC,  we  observe  that  an  element  as  pdd'p'  — 

dy(2ry  —  t/2)2.  .•.  APCB  =  £  area  of  the  generatrix,  =  £;rr2.  But  ABC  A'  = 
2r  X  A  A'  =  2r  X  #r  =  27Tr2.  .  •.  AC  A'  =  fzrr2,  or  the  entire  area  of  the 
cycloid  =  3?rr2.  Observe  that  both  these  integrals  are  to  be  taken  between  the 
same  limits,  viz.,  y  =  0,  and  y  =  2r. 


Ex.  10.  Find  the  area  of  the  curve  a4?/2  =  a262x2  — 

Area  —  f  ab. 


(6)  QUADRATURE  OF  POLAR  CURVES. 

.  Prop, — An  element  of  the  area  of  a  polar  curve  is  ^r-d0  and 
the  formula  for  the  quadrature  is     dA=-J-r2d0. 

DEM.— In  F'vj.  176  PAP'  =  dA.  But  area  PAP'  =  AP'  X  i>PD  = 
(r  -j-  dr)^rdO  =  ^r"dO,  omitting  ^rdrdO,  and  also  remembering  that  P  D  =  rdQ. 
.  • .  dA  =  ir2dO.  Q.  E.  D. 

Ex.  1.  Find  the  area  of  the  spiral  of  Archimedes. 
SOLUTION.  — The  equation  is  r  =  -— 0  ;    whence  dr  = 

mf£ 

—- (Z9,  and  dO  =  27tdr.    Substituting  in  the  formula,  we  have 

/7T 

dA  =  Ttr^dr,  and  A  =  \itr*  -f  C.  If  we  estimate  the  area 
from  the  pole,  we  have  4  =  0,  when  r  =  0,  and  conse 
quently  (7=0.  The  corrected  integral  is,  therefore,  A  =  FIG-  181 . 
ijrr3.  This  is  the  general  expression  for  the  area  passed  over  by  the  radius  vector  in 
its  revolution  from  its  starting  at  0  to  any  value,  as  r.  At  the  end  of  one  revolution 
r  =  AP  =  1,  and  A  =  area  of  the  first  spire  =  AmP  =  iir  =  i  the  area  of  a 
circle  described  with  A  P  as  a  radius.  At  the  end  of  the  second  revolution  r  =  2, 
and  the  area  traced  by  the  radius  vector,  or  A}  =  £#.  This  evidently  includes 
twice  the  first  spire  -f-  the  second.  The  area  of  the  second  spire  is,  therefore 
£  if  —  |  or  =  2;r.  The  area  of  the  first  two  spires  is  \Tt. 

Ex.  2.  Show  that  the  area  of  the  Napierian  logarithmic  spiral  is  ^ 
the  square  described  on  the  radius  vector. 

Ex.  3.  Find  the  area  of  the  Lemniscate  of  Bernouilli.   (r9  =  a-cos  20. ) 

SOLUTION. — From   the  formula  we  have  dA  =  ^a2cos20d0.      "Whence   A    = 

/**" 
|a«  I     cos  20  dO  =  ia'sin  20  =  4a2.     .  • .  The  entire  area  is  a2,  i.  e.,  the  square  on 

-V 

the  semi-axis. 

Ex.  4.  Trace  the  curve  r  =  a(cos  10  -f  sin  20),  and  find  its  area. 


QUADRATURE  OF  SURFACES  OF  REVOLUTION.        169 

SUG'S.  r"  =  a2(cos2  20  +  sin2  20+2  sin  20  cos  20)  =  a'2(l  +  sin  40)  .  • .  A  = 
J*|oa(l  +  sin40)dG  =  £«2{/dO  +  /sin40dO}.  The  entire  area,  which  is  com 
prised  between  0  =  0  and  6  =  27T,  is  jra*. 


SECTION  XL 

Quadrature  of  Surfaces  of  Eevolution, 

252.  DBF.—  JL  Surface  of  Revolution  is  a  surface  gener 
ated  by  a  line  (right  or  curved)  revolving  around  a  fixed  right  line  as 
an  axis,  so  that  sections  of  the  volume  generated  made  by  a  plane 
perpendicular  to  the  axis  are  circles. 

ILL'S.  —  A  right  line  parallel  to  and  revolving  around  another  right  line,  in  the 
manner  described  in  the  definition,  generates  a  cylindrical  surface  with  a  circular 
base.  A  semi-ellipse  revolving  around  its  transverse  axis  generates  a  prolate 
spheroid,  around  its  conjugate  axis  an  oblate  spheroid.  These  two  are  varieties  of 
ellipsoids.  A  paraboloid  is  generated  by  the  revolution  of  a  parabola  around  its 
axis.  The  number  of  these  surfaces  is,  of  course,  infinite  ;  and  the  specific  char 
acter  of  each  depends  upon  the  nature  of  the  generating  curve. 

.  Prop.—  The  differential  element  of  a  surface  of  revolution  is 
dS—  InVdy*  +  dx\ 


DEM.—  Let  M  N  be  the  generatrix,  AX  the  axis  of  rev- 
olution  and  PP'  two  consecutive  points  on  the  generatrix. 
As  M  N  revolves  about  AX,  any  point,  as  P,  whose  co 
ordinates  are  x  and  y  describes  a  circle  whose  circumference 
is  2  Try,  The  circumference  described  by  the  consecutive 
point  P'  will  be  Zit(y  +  dy).  PP'  describes  the  frustum 
of  a  cone  whose  surface  is  half  the  sum  of  the  circumfer-  -pIG  132. 

ences   traced   by    P   and    P'    multiplied  by    PP',  or  = 
2  Try  +  2ity  +  2rtdy  ^dtftj^_d^     whence  omitting  Zitdy,  it  being  an  infinitesimal 

of  a  higher  degree  than  the  other  terms,  we  have  dS=  27ty\/dy*-\-dxz.     Q.  E.  D. 

Son.—  To  apply  this  formula,  let  y  =  <p(x)  be  the  equation  of  the  genera 
trix,  from  which  find  dx  or  dy,  and,  having  substituted  in  the  formula, 
integrate  between  the  proper  limits. 

Ex.  1.  Find  the  surface  of  a  sphere. 

SOLUTION.  —From  y*  +  #2  =  r2  we  have  dy*  =  ^—  -.     .'.  dS=  2*vjdas«  H  —  — 

=  Znrdx  ;  and  S  =    j    Zrtrdx  =  %itr\     This  being  the  surface  of  a  hemisphere 
the  entire  surface  of  the  sphere  is  ±7tr'2. 


170  PBOPERTIES  OF  PLANE  LOCI. 

Con.  1. — Since  ;rr8=  area  of  a  great  circle,  the 
surface  of  a  sphere  ===  4  great  circles.     Again,  since 
4?rr2  =  2/rr  x  2r,  #ie  surface  of  a  sphere  = 
circumference  of  a  great  circle  X  </ie  diameter. 

254.  COR.  2. — JTie  portion  of  the  surface  gen 
erated  by  any  arc,  as  PS  is  a  zone.     To  find  the  surface  of  a  zone,  let 
A  D  =  a  and  A  E  =  b  and  integrate  dS  =  2;rrdx  between  these  limits , 


thus 


•,  S  =     /  2/rrdx  =  2;rr(b  —  a)  ;    i.  e.}  the  surface  of  a  zone  —  its 

altitude  multiplied  by  the  circumference  of  the  great  circle  of  (he  itphere. 
Hence,  also,  the  surfaces  of  zones  on  the  same  sphere  are  to  each  other 
as  their  altitudes. 

SCH.  —  If  a  cylinder  be  circumscribed  about  a  sphere,  its  convex  surface 
is  the  circumference  of  its  base  (27fr)  X  by  its  altitude  (2r)  ;  i.  e.,  4?rr2, 
the  same  as  the  surface  of  the  sphere.  Add  to  this  2?rr2,  the  upper  and 
lower  bases,  and  the  entire  surface  of  the  cylinder  is  found  to  be  6?rr2,  The 
surface  of  the  cylinder  is,  therefore,  to  the  surface  of  the  sphere  as  3  to  2. 
This  fact,  together  with  the  same  relation  between  the  volumes,  was  dis 
covered  by  Archimedes,  and  the  discovery  was  so  much  thought  of  by  him 
that  he  expressed  a  wish  that  the  device  on  his  tombstone  might  be  a  sphere 
inscribed  in  a  cylinder.  The  great  geometer  was  murdered  by  the  soldiers 
of  Marcellus,  B.  C.  212,  though  contrary  to  the  orders  of  that  general. 
Marcellus  executed  the  device  of  the  sphere  and  cylinder  upon  the  tomb, 
and  buried  the  philosopher  with  honors.  140  years  after,  when  Cicero  was 
questor  in  Sicily,  the  place  of  the  grave  had  been  forgotten  ;  but  he,  chanc 
ing  to  remember  the  device  upon  the  stone,  sought  out  and  restored  the 
tomb,  which  had  become  overrun  with  weeds  and  thorns. 

Ex.  2.  To  find  the  surface  of  the  paraboloid. 

SOLUTION.  —  From  y2  =  2px,  we  have  dx*  =  I—2L.     ...  dS  =  —y(pz  -f-  y*)  <fc 

and  8  =s    C^-W  +  *rfy  =       (P2  +  lf~}*  +  C.     Beckoning  S  from  y  =  0, 


C  =  —  f  ftp*  and  the  corrected  integral  is  S  =      [(p2  -f  y     —  j>3],  which  may  be 
satisfied  for  any  value  of  y. 

Ex.  3.  To  find  the  surface  generated  by  the  revolution  of  the  hypo- 
cycloid  x*  -f-  T/3"  =  c^  about  the  axis  of  x. 

S  =  2arc*  /  y*  dy  —%Tt&  ;   whence  2S  =  —  ,  the  area  of  the 

fixed  circle. 
Ex.  4.  Find  the  surface  generated  by  the  cycloid  revolved  about  its 


Suo's.     dS  =  2?r(2r)  v(2r  —  ?/)~*dv,  which    integrated  hv   fnrmnU    A  i 


CUBATURE  OF  VOLUMES  OF  REVOLUTION.          171 

lr  _  yf  _  -L/7rrv/2r(2r  —  y)*  -f  C.  This  taken  between  y  =  2r,  and 
y  =  0  gives  for  half  the  surface  ^f-Ttr2,  and  for  the  whole  surface  fi/"  times  the  gen 
erating  circle. 

Ex.  5.  Find  the  surface  of  the  prolate  spheroid. 

SOLUTION.— From    a2?/2   -f-  llW  —  asb2    we    get    dy*  =  —  -dx?.       .-.    <LS  = 

(<141/*  -f-   &4X2\2-                  2?ir                                                           i-                27T&  i 

— • )  dx  =   — \ct  (ct^62  —  b*x2)  -j-  54x2]  dx  "=• (ct1*  —  O!2e2x2)  dx  =^ 
a4?/-       /a'2                                                        a2 

27T-(a2  —  e-x2)*dx.    Whence  half  the  surface  of  the  ellipsoid  =  27T-  /  (a2  — e2x2)  *dx. 
To  integrate  (a2  —  e2#2)*dx  apply  formula  <JJf  which  gives  J"(a2  —  e2x2)*dx  = 


(a2  —  e2x2)* 


Beckoning  S  =  0  for  x  =  0,  (7=0;  and  then  putting  x  =  a  we  have  for  half  the 
surface  itab[(l  —  e2)*  +  -sin-'c],  or  HW  -j sin-'c. 


Ex.  G.  Apply  the  above  formula  to  find  the   surface  of  the  prolate 
spheroid  whose  generatrix  is  25t/«  -f  16x*  =-  400. 

Area,  235.41  nearly. 


SECTION  XII. 
Oubature  of  Volumes  of  Eevolution, 

.  Prop. — The  differential  element  of  volume  of  a  solid  of  revo 
lution  is  dV  =  7ry2dx. 

DEM.— Using  Fig.  182,  with  the  same  notation,  the  volume  generated  by  the  revolu 
tion  of  D  PP  D'  about  D  D',  is  a  frustum  of  a  cone,  and  is  therefore  equal  to  three 
cones  having  for  their  common  altitude  the  altitude  of  the  frustum  (dx),  and  for  bases 
the  upper  base  (ny2),  the  lower  base  [flr(y  +  c?i/)a],  and  a  mean  proportional  between 


the  two  bases  [Tty(y  -f-  dy}~\.     Hence  dT=  — 

(omitting  terms  having  infinitesimals  of  a  higher  order)  Tty^dx.     Q.  E.  D. 

Ex.  1.  To  find  the  volume  of  a  sphere. 

SOLUTION,     y^  =  r2  —  #-.     .  • .  d  V  =  7T(r2  —  x2)(ix  =  nr^dx  —  itx^dx,  V  — 
=  f  itf* ;  or,  letting  D  =  the  diameter,  and  doubling  for  the 
entire  volume  V  —  ^ 


172  PROPERTIES  OF  PLANE  LOCI. 

2  56*  COR.  1.  —  Since  |7fr3  =  ^rx  4^rs,  the  volume  of  a  sphere  =  the 
surface  x  ^  the  radius. 

257*  COR.  2.  —  The  volumes  of  spheres  are  to  each  other  as  the  cubes 
of  their  radii  or  their  diameters. 

258.  COR.  3.  —  The  portion  of  a  sphere  included  between  two  parallel 
planes  is  called  a  segment.  If  a  and  b  represent  the  distances  of  these 
planes  from  the  centre,  the  volume  of  the  segment  is 

V  =  7r[r*(b  —  a)  —  J(b»  —  a')]. 


50.  COR.  4.  f  Trr3  =  %  of  m*  x  2r  =  •§•  of  the  circumscribed  cylinder. 
See  Scholium  under  ART.  254. 

Ex.  2.  Find  the  volume  of  the  Prolate  Spheroid.  Also  of  the  ob 
late  Spheroid.  Show  that  each  is  f  of  the  circumscribed  cylinder. 
Deduce  from  each  the  volume  of  the  sphere. 

Ex.  3.  Find  the  volume  of  the  paraboloid. 

Volume  =  one  half  the  circumscribed  cylinder. 

Ex.  4.  Find  the  volume  of  the  cissoid  revolved  about  its  asymptote. 

SOLUTION.  —  Let  the  curve  AM  revolve  about  B"T, 
then  will  P  D  P  '  D  '  be  an  elemental  section  of  the  vol 
ume  if  DD'  =  dy.  PD  =  2a  —  a,  and  the  circle 
traced  by  P  D  in  its  revolution  is  Tt(1a  —  a;)2.  There 
fore  the  element  of  volume  or  d  V=  7t(2a  —  x)ydy.  From 

x^ 

the  equation  of  the  curve  v2  =  -  ,  we  have  dy  = 

2«  —  x 


J^te  =  22      f  ^"^   ""  dx.     Whence  sub-  FIG.  184. 

i  1  f          •     |  4 

stituting,  dV  =  7T(3«  — ft)(2cp;  —  aft)  ax,  =  Sartx  (2a  —  x)  dx  —  Ttx  (2a  —  x)  dx. 


These  terms  may  be  integrated  by  the  use  of  formulas  ^  and  <^f,  and  we  find 

JL 
a  —  xYdx  =  TtW. 


=    /3a7rx*(2a  —  xfidx—  J    7 


SUCTION  XIII. 

Equations  of  Curves  deduced  by  the  aid  of  the  Calculus, 

200.  DEF. — The  Tractrix  or  Equitangential  Curve  is  generated 
by  the  motion  of  a  weight  drawn  upon  a  plane  by  a  cord  of  constant 
length,  the  extremity  of  the  cord  moving  along  a  straight  line  not.  in 
the  direction  of  the  cord  itself.  The  portion  of  the  tangent  inter- 


EQUATIONS  OF  CURYES  DEDUCED  BY  THE  AID  OF  THE  CALCULUS.    173 

cepted  between  the  curve  and  the  fixed  line,  is  a  constant  quantity 
(the  length  of  the  string),  and  hence  the  second  appellation. 

ILLUS. — Let  a  weight  be  placed  at  B,  Fiy.  185,  with        y 
a  string  of  the  length  A  B  attached.     As  the  extrem 
ity  of  the  string,  A,  is  carried  along  the  line  AX,  B 
will  trace  BM,  the  tractrix.     (This  conception  sup 
poses  friction  to  exist  but  not  momentum.)     "When 
the  extremity  of  the  string  is  at  any  point  in  the  line,     ~A     D  D' 
as  T,  the  weight  will  be  at  P,  and  it  is  evident  that  FIG.  185. 

PT,  the  tangent,  will  be  constant. 

26 1.  Prob.—Tofind  the  equation  of  the  Tractrix. 

SOLUTION.— Let  P  and  P'  be  consecutive  points  on  the  curve,  PD  =  y, 
PE  =  —  dy,  (minus  since  y  decreases  as  x  increases),  A  D  =  x  and  D  D'  =  dx. 
Let  AB  =  PT  =  a.  Then  DT  =  v/a*  —  y2,  and  PE  :  EP'  : :  PD  :  DT, 
or  by  the  notation  —  dy  :  dx  : :  y  :  (a2  —  y2)  .  .  • .  -^  -j =  0. 

Of  this  differential  equation  may  be  obtained  from  the  general  differential  value 
of  the  tangent ;  thus  (l  -f  ttjV****  whence  ^  =  ± ^ -.  The  -f-  sign 

indicates  that  the  curve  is  generated  by  motion  to  the  left,  as  then  tan—'  --  is  -f-, 

dx 

and  the  —  sign  is  to  be  taken  when  the  curve  is  generated  as  in  the  illustration  above. 
In  order  to  have  the  equation  in  finite  terms  it  remains  to  integrate  this  differential 
between  proper  limits.  The  inferior  limit  of  x  is  evidently  0,  to  which  y  =  a  cor 
responds.  The  superior  limit  must  be  left  general,  as  the  curve  extends  to  infinity. 

2  2  ^  /•* 

Putting   the   equation   in    the   form  dx  = — - — dy  we  have      /   dx  =. 

y  %/o 


_ 

J 


.  The  equation  is  x  =  a  log  a  + 


262.  Prob. — To  find  the  equation  of  a  curve  whose  subnormal  is 
constant. 

SOLUTION. — The  general  differential  value  of  the  subnormal  is  y-~.     .•.  Letting 

p  be  the  constant  value  of  the  subnormal,  we  have  y~=p,  or  y  dy  =  p  dx.  In 
tegrating,  y2  =  2px  -f-  C.  This  is  the  equation  of  the  parabola,  as  it  should  be, 
since  the  constant  subnormal  characterizes  that  curve.  C  is  not  determined  by 
the  problem.  If  the  condition  "which  passes  through  the  origin,"  were  added 
to  the  problem,  C  would  be  0. 


174  PROPERTIES  OF  PLANE  LOCI. 

263.  Prob. — To  find  the  equation  of  the  curve  whose  normal  is  of 
constant  length. 

SOLUTION. — The  general  differential  value  of  the  normal  is  yfl  -f-  —  V,      Put 
ting  this  equal  to  the  constant  r,  we  have  y(  I  -4-  —  V  =  r,  or  —  = 1,  or 

\         etcv  dx*       «/* 

dx  = : — j_  =  ?/(r2  —  ys)    dy.     Placing  the  origin  on  the  curve,  so  that  thp 

^2  __    y2) 2 

inferior   limit  of  the   integral    shall   be  x  =  0,  y  =  0  and  we  have    /    dx 


f 

«/0 


y(r2  —  y2)  *dy,  or  a  =  r  —  (r*  —  y*f  whence  y«  =  2rx  —  z2.      This  is  the 


well  known  equation  of  the  circle,  as  it  should  be,  the  normal  of  the  circle  being 
its  radius,  and  hence  constant  : 


,  frobm  —  To  determine  the  equation  of  the  curve  whose  subtan- 
gent  is  constant  (m). 

m  log  y  =  x  +  C.     The  logarithmic  curve. 

265.  Prob.  —  To   determine  the  equation  of  the  curve  whose  sub 
normal  varies  as  the  square  of  the  abscissa. 

y2  =  f  x3  +  C.     The  semicubical  parabola. 


266.  Prob. — To  determine  the  equation  of  a  curve  such  that  the 
area  shall  equal  twice  the  product  of  its  co-ordinates. 

SOLUTION,     fydx  =  2xy,  or  ydx  =  2xdy  +  Zydx  or  --  =  —  £-'-.     Integrating, 

y  % 

L  11 

log  y  =  —  £  log  x  -f  C=  —  log  x*  -f-  C.     .'.   C=  log  z*y.     As  log  x2y  is  constant 

sry  must  be  constant ;   therefore  we  may  put  x'zy  =  m,  or  an/2  =  m2  is  the  equa 
tion  sought. 


267.  Prob.  —  Find  the  equation  of  the  curve  whose  arc  varies  as  the 
square  root  of  the  third  power  of  the  abscissa. 

SOLUTION,    ^(dx^-^-dy12)    =  mo;2,  using  m  for  any  constant  factor.     Removing 


the  sign  of  integration  by  differentiation  and  squaring  both  sides  we  have  dx*-\-dy* 
=  f  vrfxdx*,  or  dy  =  (fmsa;  —  1/do^  Whence  integrating,  y  ==          fi™^  —  1  )* 
the  semi-cubical  parabola. 


OF  TANGENTS  AND  NORMALS.  175 


SECTION  XIV. 

Of  Tangents  and  Normals, 

[WITHOUT     THE    AID    OF    THE    CALCULUS.] 

[NOTE.— Students  taking  a  shorter  course,  without  the  Calculus,  will  omit  the  thirteen  preceding 
sections  of  this  chapter,  and  conclude  the  course  with  this  and  the  following  section.  Such  a» 
take  the  fuller  course  will  not  need  to  read  this  section,  but  should  read  Sec.  XV.] 

268     J?rob» — To  produce  the  equation  of  a  tangent  to  a  plane  curve. 

SOLUTION. — Let  M  N  be  any  plane  curve  whose 
equation  is  y  — /(x)*,  or /(a;,  y)  =  0.  Let  P'be  a 
point  in  the  curve  whose  co-ordinates  are  (x',  y'}, 
and  P"  a  point  whose  co-ordinates  are  (x",  y"}. 
Then  the  equation  of  the  secant  line  RT  is 

y  —  y'  =  y—^—(x  —  x)  (31).     Now  as  P'  and 

P    are  points  in  the  curve  their  co-ordinates  will 
satisfy  the  equation  of  the  curve,  and  we  have 

(1)  T/' =/(#')  and  2/"=/(x");    or 

(2)  f(x;  T/')=  0  and  /(x",  y")  =  0. 

If  now  we  can  find  the  value  of  — — -  from  the  two  equations  (1)  or  (2),  in 

x  —  x 

such  a  form  that  it  will  take  a  determinate,  finite  form  when  y'  =  y",  and  x'  =  x", 
that  is  when  P'  and  P"  coincide,  this  value  substituted  in  the  equation  of  the 
secant  will  transform  it  into  the  equation  of  a  tangent,  since  it  is  evident  that  as 
P'  and  P"  approach  each  other  the  line  RT  passing  through  them  approaches  a 
tangent,  and  becomes  a  tangent  when  the  points  coincide.  Q.  E.  D. 

Ex.  1.  Produce  the  equation  of  a  tangent  to  the  parabola. 
SOLUTION. — Letting  (x',  y')  and  (x"  y")  be  two  points  on  the  curve,  the  equa 
tion  of  a  line  passing  through  them  is  y  —  y'  =  -, --(x  —  x'),  which  is  the 

equation  of  a  secant,  since  the  two  points  are  in  the  curve.  Moreover  as  (x',  y) 
and  (x",  y"}  are  in  the  curve  they  satisfy  the  equation  of  the  curve  y2  =  2px, 
giving  (1)  y"2  =  2px',  and  (2)  y"2  =  2px".  Subtracting  (2)  from  (1)  y'2  —  y'"2  = 
2p;x'  —  x"),  or  dividing  by  x'  —  x",  and  y'  -\-  y",  we  have 

y. -L-  —  £. —   =  £  when  the  point  (x",  y")  is  made  to  coincide  with 

»'  -  **     V  +  y"      y 

(x',  y').  Substituting  this  value  in  the  equation  of  the  secant,  we  have  for  the 
tangent  y  —  y'  =  ^(x  —  x'),  or  yy'  —  y'2  =  px  —  px',  or  yy'  =  y'*  -f  px—px'. 

*  This  is  read  "  y  equals  a  function  of  x,"  and  is  simply  a  general  form  comprehending  every 
equation  containing  x  and  y,  supposed  solved  in  respect  to  y.  The  second  form,  read  "  function 
ol  z  and  y  —  0,"  indicates  the  same  thing  except  that  the  equation  is  not  supposed  to  be  solved. 


176  PBOPEBTIES   OF   PLANE  LOCI, 

But  as  (#',  y')  is  a  point  in  the  curve  y'2  =  %px  .     Substituting  this,  and  uniting 
terms  we  have  yy'  —  p(x  -f-  x'). 

Ex.  2.  Produce  the  equation  of  a  tangent  to  the  parabola  whose 
parameter  is  9,  at  x  =  4.  Construct  the  tangent  from  its  equation, 
and  then  construct  the  parabola,  thus  observing  that  the  right  line  is 
tangent  to  the  curve  at  the  given  point. 

SOLUTION. — [We  might  proceed  as  in  the  general  method,  but  it  will  be  better  to 
use  the  equation  of  the  tangent  to  the  parabola,  both  because  it  will  require  less 
work,  and  because  it  is  important  that  this  form  should  be  made  familiar.] 

The  equation  of  this  parabola  is  y:  =  9x.  For  x  =  4,  y  ==  ±  6.  Taking  the  point 
(4,  6)  we  have  x'  =4,  and  y'  =6.  Hence  yy'  =  p(x-\-x')  becomes  &y=.ty(x  -\-  4), 
or  y  =  %x  4-  3,  as  the  equation  of  a  straight  line  which  is  tangent  to  the  parabola 
y*  =  9#  at  x  =  4,  y  =  6. 

Construction. — 1st  Constructing  the  line  y  =  %x  -f-  3, 
we  find  that  it  cuts  the  axis  of  y  at  (0,  3) ,  that  is  at  G, 
and  the  axis  of  x  at  ( —  4,  0),  that  is  at  "T.  Whence  we 
draw  RT.  2nd.  Constructing  the  parabola  y-  =  9ic, 
we  find  that  the  line  RT  touches  it  at  P,  whose  co 
ordinates  are  (4,  6). 

As  x  =  4  gives  y=  ±  6,  we  have  another  point  (4,  — 6) 
which  is  embraced  by  the  example.  The  equation  of  a 
tangent  at  this  point  is  y  =  —  |«  —  3.  [Let  the  student 
produce  and  illustrate  it.]  FIG.  187. 

Ex.  3.  Produce  the  equation  of  a  tangent  to  the  parabola  whose 
parameter  is  10,  at  x  =  10,  and  construct  and  illustrate  as  above. 


Ex.  4.  Produce  the  equation  of  the  tangent  to  the  ellipse  referred 
to  its  axes. 

Suo's.—  Equation  of  secant,  y  —  y'  =  y  ,  ~~  y-(x  —  x).     The  equation  of  the 

x  —  x 

ellipse  A*y*  -j-  B*x*  =  A*B*,  satisfied  for  the  points  (x,  y')  and  (x",  y"},  gives 

(1)  A*y'*    +  B*x'*    =  A*B*  and 

(2)  Ay*  -f  jgy-2 


Subtracting,         A*(y'*  —  y"^  +  B*(x'  *  —  x"*)  —  0  ;  whence 
y'  —y"  B  2/x'  -f-  x"  \  B*x' 

=  ~        '  when  (x  •  y  )  becomes  ^incident 


with  (x',  y').     Hence  the  equation  of  the  tangent  is 

J?2«' 

y  —  J/*  =  —  ^pjrH*  —  x  ')  »  or>  clearing  of  fractions  and  transposing, 
A*yy'  +  B*xx'  =  A*y'*  -f-  B*x*.     But  since  (x  ',  y')  is  in  the  curve  A*y'*  -f  B*x'*  = 
which  value  substituted  gives  for  the  equation  of  th*  tangent  to  the  ellipse 
-f- 


TANGENTS.  177 

Ex.  5.  What  is  the  equation  of  a  tangent  to  the  ellipse  whose  axes 
are  6  and  4,  at  x  —  2  ?     Construct  as  above. 

Ans.,   Calling   the   point  of    tangency    (2,    1.49)   the   equation  is 

y  =  —  .5966tf  +  2.66.     Calling  the  point  of  tangency  (2,  —  1.49) 

the  equation  is  y  =  .596607  —  2.66. 

Ex.  6.  Produce  the  equation  of  a  tangent  to  3y2  -f  x2  =  5,  at  x  =  1. 
Construct  and  illustrate  as  before. 

SUG.—  See  Ex.  2,  page  97,  second  solution. 

Ex.  7.  Show  that  the  form  of  the  equation  of  a  tangent  to  a  circle 
at  a  given  point  (x',  y'}  in  the  circumference  is  yy'  +  xx'  =  J?2. 

Ex.  8.  What  is  the  equation  of  a  tangent  to  a  circle  whose  radius 
is  5  at  x  =  3  ?  Equation,  y  =  qp  f  x  =h  6£. 

Ex.  9.  What  is  the  equation  of  a  tangent  to  a  circle  whose  radius 
islOat#  =  —  3?    Ati/=  —  4?    Aty  =  l? 

One  equation  is  y  =  ±  £  v2Ix  —  25. 


Ex.  10.  Produce  the  equation  of  the  tangent  to  the  hyperbola. 

Equation,  A*yy'  —  B2xx'  =  —  A^B*. 

Ex.  11.  Produce  the  equation  of  a  tangent  to  the  hyperbola  whose 
axes  are  4  and  2,  at  x  ==  4.     Construct  and  illustrate. 

The  points  of  tangency  are  (4,  \/3),  and  (4,  —  v/3). 
Equation  of  tangent  at  (4,  v/3),  y  =  Jv^3#  —  Jv/3. 

(4  —  \/3),  y  =  —  JV/&P  +  £\/3. 

Ex.  12.  Produce  the  equation  of  a  tangent  to  3i/s  —  2#2  =  10, 
at  x  =  4.     Is  there  more  than  one  tangent  ?     Construct  as  above. 

Equation,  y  =  db  .7127^  ±  .8909. 


Ex.  13.  Produce  the  equation  of  a  tangent  to  y9  =  -— — ,  at  x  =  2. 

SUG'S. — Reasoning  as  on  Ex.  1,   we  have  y'a  = ;,  and  y"z  =  —^ -. 

4  —  x  4  —  x, 

_,  ..       ..  x'3(4  — x")— a;"3(4  — x')      4x'3  —  x'*x"  —  4x"3  -fx"-'x' 

Subtractog,  r-jT:       _^_^__  -(4_,.)(4_^.) 

— .     Whence  dividing  by  y'  -{-  y",  and  x  —  x",  we 


- 

(4  —  a;  )(4  —  x   ) 

u'—v"       4(x'24-xV  -f  x"2)  —  x'x"(x'  4-x")       Gx'*  —  x"*     , 
have  -  -  --  =  -  1  -  -  -  --  rr-^  -  =  -r-t  -  r-  when  x    =x  , 
x'  —  x"  (y1  4-  ?/  ")(4  —  x')(±  —  x   )  2/(4  —  x  j* 


and  y"  —  y'  .    Substituting  this  value  of  '—  -  -r,  in  the  general  equation  for  the 

j*     -  Q* 


178  PROPERTIES  OF  PLANE  LOCI. 

secant,  we  have  for  the  general  equation  of  a  tangent  to  this  locus  ?/  —  y    = 

— "r^(x  —  x  )•     For  the  point  on  the  curve  x  =  2,  this  gives  for  the  two 

tangents  y  =.  %x  —  2,  and  y  =  —  2x  -j-  2. 


200.  COR. — The  tangent  of  the  angle  which  a  tangent  to  an  ellipse  at 
(x,  y)  makes  with  the  axis  of  abscissas  is  —  —  -,  to  an  hyperbola,  — -,  and 

to  a  parabola  -. 

y 

DEM.  — These  are  the  values  of  the  coefficient  — '-—  as  obtained  in  the  several 

x  —  x 

cases  in  Ex.  4,  10,  and  1.     But  this  coefficient  is  the  tangent  of  the  angle  which  the 
line  makes  with  the  axis  of  x  (31,  COB.  1). 

Ex.  1.  "What  angle  does  a  tangent  to  ?/2  —  Sx  make  with  the  axis  of 
x,  when  the  point  of  tangency  is  at  x  =  2  ?     At  x  =  5  ?     At  x  =  10  ? 
Answers,  Tan^l  or  45°,  tan-\— 1)  or  135°  ;   tan"1  (  ±.63245)  or 

32°  18'  30"  and  147°  41'  30" ;    tan"1  (db  .447214)  or  24°  5'  41" 

and  155°  54'  19". 

Ex.  2.  What  angle  does  the  focal  tangent  to  the  parabola  make 
with  the  axis  of  #? 

Ex.  3.  In  an  ellipse  whose  axes  are  10  and  6  what  angle  does  a 
tangent  at  the  point  x  =  2  make  with  the  axis  of  x  ? 

Am.,  165°  19'  33"  and  14°  40'  27". 

27O.  Son.  — To  determine  at  what  point  on  a  given  curve  a  tangent  must 
be  drawn  to  make  a  given  angle  with  the  axis  of  x ;  or,  what  is  the  samo 
thing,  to  find  a  point  at  which  a  curve  has  a  given  direction,  put  the  gen 
eral  value  of  the  co-efficient  of  x  in  the  equation  of  the  tangent  to  the  par 
ticular  curve  equal  to  the  tangent  of  the  proposed  angle.  The  equation 
thus  formed  together  with  the  equation  of  the  locus  will  enable  us  to  find  the 
values  of  x  and  y  which  locate  the  point  sought.  In  case  the  tangent  is  to  be 
parallel,  the  co-efficient  is  put  equal  to  0 ;  if  perpendicular,  equal  to  oo. 

Ex.  4.  At  what  point  on  an  ellipse  whose  axes  are  16  and  8  must  a 
tangent  be  drawn  to  make  an  angle  with  the  axis  of  x  whose  tangent 
is  2  ?  At  what  point  to  make  an  angle  of  45°  ?  At  what  point  to 
make  an  angle  of  135°  ?  At  what  point  is  the  tangent  parallel?  At 
what  point  perpendicular  ? 

Suo's. — The  general  value  of  the  tangent  of  the  angle  which  a  tangent  to  an 
ellipse  makes  with  the  axis  of  x  is  —  - — .  Hence  for  the  1st  inquiry  we  have 

-r  ~  =  2,  and  64y*  -{-  16z*  =  1024,  from  which  to  find  x  and  y. 


SUBTANGENTS.  179 

Ex.  5.  At  what  point  on  an  hyperbola  whose  axes  are  8  and  6  must 
a  tangent  be  drawn  to  make  an  angle  of  45°  with  the  axis  of  or? 
What  angle  does  the  focal  tangent  make  with  the  axis  of  x  ? 

Ex.  6.  On  an  elliptical  track  whose  transverse  axis  is  1  mile,  and 
whose  conjugate  is  f  of  a  mile  in  length  and  coincides  with  the  me 
ridian,  in  what  direction  is  a  man  travelling  when  he  is  on  the  north 
west  quarter  of  the  track,  travelling  around  from  left  to  right,  and  is 
at  40  rods  from  the  transverse  axis  ? 

Answer,  North  25°  14'  21"  east. 


SUBTANGENTS. 

271.  DEF.  —  A.  Subtangent  is  the  portion  of  the  axis  of  ab 
scissas  intercepted  between  the  foot  of  the  ordi- 

nate  from  the  point  of  tangency,  and  the  inter 

section  of  the  tangent  with  this  axis  ;  or  it  may 

be  denned  as  the  projection  of  the  correspond 

ing  portion  of  the  tangent  upon  the  axis  of  x.    —  ~~7^~ 

DT  is  the   sub  tangent   corresponding   to   the     / 

point  P. 

272.  f  rob.—  To  find  the  length  of  the  sub-  FIG  188 
tangent. 

SOLUTION.  —  Letting  a  represent  the  angle  which  a  tangent  to  the  curve  at  the 
specified  point  makes  with  the  axis  of  x,  find  tan  a  as  in  (268).     Whence  from  the 

triangle  P  DT",  Fig.  188,  we  have  —  —  =  tan  a,  or  DT  (the  sub  tangent)  = 


A  slightly  different  method  of  solution  is  to  produce  the  equation  of  the  tangent 
to  the  curve,  and  then  find  where  it  intersects  the  axis  of  x.  This  intercept,  as 
AT,  Fig.  188,  and  the  abscissa  A  D  make  known  the  subtangent,  which  is  always 
their  algebraic  difference. 

Ex.  1.  Find  the  value  of  the  subtangent  of  the  common  parabola 
at  (x',  y'). 

SOLUTION.—  We  find  in  (268  Ex.  1)  that  tan  a  —  -,.     Whence  Subt.  =     ~  = 


273.  Sen.  —  From  this  example  we  learn  that  the  subtangent  in  a  para 
bola  is  always  equal  to  twice  the  abscissa  of  the  point  of  tangency.  Hence 
to  draw  a  tangent  to  a  parabola  at  any  point  as  P,  Fig.  188,  let  fall  the  ordi- 
nate  PD,  take  AT  —  AD,  and  draw  PT. 


180 


PROPERTIES   OF   PLANE   LOCI. 


Ex.  2.  What  is  the  subtangent  to  ?/2  =  10.r,  at  x  =  6  ?     At  y  =  8  ? 
At  y  =  12  ?     Draw  these  tangents  according  to  the  scholium. 


Ex.  3.  Find  the  value  of  the  subtangent  to  the  ellipse  at  (x't  y'). 

SOLUTION.— We  have  tan  a  = — ;  by  EC.  4  (268),  whence  Subt.  =  -Ju~  = 

A*y  tan  a 

A*y"*       A2  —  x"1 
—  •  na   -  = -, neglecting  the  —  sign  as  we  are  inquiring  simply  for  a  value, 

not  a  direction. 


prrr 


FIG.  189. 


Sen.  1. — From  this  example  we 
learn  that  the  subtangent  in  the  ellipse 
does  not  depend  upon  the  conjugate 
axis,  but  only  on  the  transverse  axis 
and  the  abscissa  of  the  point  of  tan-  C| 
gency.  Hence  if  several  ellipses  be 
drawn  on  the  same  transverse  axis  the 
subtangents  corresponding  to  the  same 
abscissa  are  equal.  Thus  in  Fig.  189 
let  APB,  AP'B,  AP"B,  and  AP"  B  be 
several  ellipses  on  the  same  transverse  axis  CB,  then  AD  being  any  abscissa 
(a?'),  the  subtangent  corresponding  to  each  point  of  tangency  P,  P',  P",  P'",  is 

^2  -V2 

; .     Hence  the  tangents  at  these  several  points  cut  the  axis  of  as  at 

*c 

the  same  point  T. 

275.  SCH.  2. — This  property  affords  a  convenient  method  of  drawing  a 
tangent  to  an  ellipse  at  a  given  point  in  the  curve.      Thus  let  P  be  the 
point,  Fig.  189.     Draw  the  circle  upon  the  transverse  axis  CB,    draw  the 
ordinate  PD  and  produce  it  till  it  meets  the  circumference  of  the  circle 
in  P'",  and  then  draw  a  tangent  to  the  circle  at  P'".     T  being  the  point 
at  which  the  latter  intersects  the  axis  of  x,  is  also  the  point  at  which  all 
corresponding  tangents  to  ellipses  on  the  same  axis  intersect.     Hence  draw 
ing  PT  it  is  the  tangent  sought. 

276.  COR.  — The  expression  for  the  subtangent  being  independent  of 
B,  the  property  is  the  same  in  the  hyperbola  as  in  the  ellipse  (00}.     How- 

x's A2 

ever  as  x  ^>  A  in  the  hyperbola,  we  may  write  Subt.  = , — ,  in  order 

to  have  it  positive. 

Ex.  4.  I*TOp. — In  an  ellipse  or  hyperbola  if  from  any  point  in 
the  curve  a  tangent  and  an  ordinate  be  drawn  to  the  transverse  axis, 
half  this  axis  is  a  mean  proportional  between  the  distances  of  the 
intersections  from  the  centre. 


NORMALS. 


181 


FIG.  190. 


DEM. — Let  P  be  any  point  in  the  curve  and 
PD,  PT  the  ordinate  and  tangent.  Then 
AT  :  AB  : :  AB  :  AD.  For  we  have  the 
equation  of  a  tangent  at  P  (x',  y'),  A*yy  ± 
B-xx'  =  ±  A-B-.  Making  y  =  0,  we  get 

x—  — .     But  x  is  AT  and  x'  is  A  D.     Hence 

.  x 

x:A::A:x',  or  AT:  AB   ::  AB  :  AD. 

The  demonstration  being  the  same  in  each  case. 

277.  Sen.  1. — To  draw  a  tangent  to  a 
hyperbola  at  a  given  point.      From  the 
given  point  of  tangency  P,  Fig.  191,  let 
fall    the    ordinate   PD  ;    and  upon  the 
transverse  axis  HB,  and  the  abscissa  AD, 
draw  semi-circumferences.     From  their 
intersection  let  fall  LT  a  perpendicular 
upon  the  axis  of  x.     Draw  a  line  through 

P  and  T  and  it  is  the  tangent  sought.  FIG.  191. 

Proof.  Drawing  AL  and  LD,   we  have  AD  (or  x')   :  AL   (or  A)   : :  AL 

A* 
(or  A)  :  AT.     Whence  AT  =   —  and  is  the  intercept  made  by  a  tangent 

at  P. 

278.  Sen.  2.— The  pupil  can  scarcely  fail  to  notice  the    close  analogy 
between  the  forms  of  the  equations  of  the  conic  sections  and  the  equations 
of  their  tangents ;    by  simply    dropping  the  accents    the   latter  return  to  the 
former.     Thus  dropping  the  accents 

A*yy'  +  B*xx'  =  A*B* becomes  A*y*  + 

A2yy'  --  B*xx'  =  —  A*B* becomes  A*y*  — 

2/y'  ==^  P(X  +  x>) becomes  y*  =  2px,     and 

yy'  -{-  xx'  —  Ri becomes  y*  -f  x*  =  R*. 


NORMALS. 

270.  DEF. — A.  ^formal  to  a  plane  curve  is  a  perpendicular  to  a 
tangent  at  the  point  of  tangency. 

280.  JProb. — To  produce  the  equation  of  a  normal  to  a  plane  curve. 

DEM. — Let  PE  be  a  normal  to  the  curve 
M  N  at  the  point  P,  the  co-ordinates  of  which 
are  (x,  y'}.  The  equation  of  a  tangent  at  the 
point  P  is  y  —  y'  =  a(x  —  x'),  in  which  a  is 
the  tangent  of  the  angle  which  the  tangent  at 
P  makes  with  the  axis  of  x,  that  is  tan  STX  ; 
and  is  to  be  determined  from  the  equation  of 
the  curve  as  in  the  preceding  part  of  this  sec 
tion.  Now  the  equation  of  a  line  passing 


FIG.  192. 


182  PROPERTIES  OF  PLANE  LOCI. 

through  (x',  y')  and  perpendicular  to  the  line  y  —  y'  =  a(x  —  x"),  is  y  —  y'  =^ 
---  (x  —  x')  (30},  the  coefficient  --  being  the  negative  reciprocal  of  the  tan 
gent  of  the  angle  which  the  tangent  to  the  curve  makes  with  the  axis  of  x. 

281.  COR.  —  The  tangent  of  the  angle  ivhich  a  normal  to  a  curve  at  a 
particular  point  makes  with  the  axis  of  x  is  the  negative  reciprocal  of  the 
tangent  of  the  angle  which  is  made  by  a  tangent  to  the  curve  at  the  same 
point. 

Ex.  1.  To  find  the  equation  of  a  normal  to  an  ellipse. 

SOLUTION.  —  The  equation  of  the  tangent  to  the  ellipse  is  A  "-yy  -f-  _Z?-xx'  =  A^B'-, 
or  y  =  —  '  ,x  -|  --  7.  Now  the  equation  of  any  line  passing  through  (x',  y')  is 
y  —  y'  =  a(x  —  x')  ;  but  in  order  that  this  should  be  a  normal  to  the  ellipse  we 
must  have  a  =  •  ,  ,  the  negative  reciprocal  of  the  tangent  of  the  angle  which  the 

A^ii' 
tangent  makes  with  the  axis  of  x.     Hence  y  —  y'  =  -^~'(x  —  a?')  is  the  equation 

of  a  normal  to  the  ellipse. 

Ex.  2.   Show  that  the  equation  of  a  normal  to  an  hyperbola  is 


Ex.  3.   Show  that   the   equation  of    a  normal  to  the  parabola  is 

y  -*  =  -%(*-*")• 

Ex.  4.  Show  that  the  equation  of  a  normal  to  the  circle  is  y  =  -,#, 

%JC 

and  hence  is  the  radius. 

Ex.  5.  What  is  the  equation  of  a  normal  to  the  ellipse  whose  axes 
are  8  and  4  at  07  =  1?     At  #  =  —  1?     Atj;=3? 

One  equation  is  y  =  db  }v  7#  qc  f  v  7. 

Ex.  6.  What  is  the  equation  of  a  normal  to  the  parabola  whose 
parameter  is  9,  at  x  =  4  ?     At  x  =  9  ?     At  x  =  5|  ? 

One  equation  is  y  =  qp  2x  =b  27. 

282.  Con.  —  The  general  expressions  for  the  tangents  of  the  angles 
which  normals  to  the  conic  sections  make  with  the  axis  of  x  are  : 

A2y'  A2y' 

For  the  ellipse  ^—t,  for  the  hyperbola  —  ^-~lt 

y'  y' 

For  the  circle        —  ,,  for  the  parabola        —  —  . 


SUBNORMALS. 


183 


SUBNORMALS. 

283.  DEF. — TJie  Subnormal  is  the  projection  of  the  normal 
upon  the  axis  of  x  ;  or  it  is  the  distance  from  the  foot  of  the  ordinate 
let  fall  from  the  point  in  the  curve  to  which  the  normal  is  drawn,  to 
the  intersection  of  the  normal  with  the  axis  of  x,  as  D  E,  Fig.  193. 

Ex.  1.  Show  that  the  subnormal  in  the 


B\x' 

ellipse  is  equal  to  — — ,  and  has  the  same 

numerical  value  in  the  hyperbola. 

SUG'S. — In  case  of  the  ellipse,  let  PE  be  the     X1 
rormal  at   P   and    ED    the  subnormal.      Now 


Whence 


SN 


-         =  tan  PE  D,  or  - 

E  D  tktbnor 


Subnor  = 


Bx 


FIG.  193. 


Ex.  2.  Show  that  in  the  parabola  the  subnormal  is  constant  and 
equals  the  semi-parameter.  Show  how  this  property  may  be  used  to 
draw  a  tangent  to  a  parabola  when  the  focus  is  known. 


284:.  Prop.  —  In  the  parabola  a  line  joining  the  focus  and  the  inter 
section  of  a  tangent  with  the  axis  of  y  (a  tangent  at  the  vertex),  is  per- 
pendicular  to  the  tangent. 

DEM.  —  Let  F  be  the  focus,  P"T  a  tangent,  and 
PE  a  normal.  Join  the  intersection  L.  with  F. 
Then  is  FL  perpendicular  to  PT.  For,  since 
AT  =  AD  (.Be.  1,  272),  TL  =  LP.  Again 
TF  =  AT  +  AF  =  AD  +  ip.  Also  FE  = 
AD+DE  —  AF  =  AD-fp  —  £p=AD-Hp. 
Whence  as  T  P  and  T  E  are  bisected  by  F  L  it  is 
parallel  to  the  normal  P  E  and  hence  perpendicular 
to  the  tangent  PT.  Q.  E.  D. 


TF=FE. 


FIG.  194. 
Also  angle  PT  F  =  T  P  F,  and 


285.  COR.      PF 
FPE=FEP. 

286.  Sen.  1.  —  Having  given  the  curve   and  its  axis,  to  find  the  focus 
draw  a  tangent  to  any  point,  as  P,  by  (273),  and  then  erect  a  perpendicular 
to  it  where  it  intersects  the  tangent  at  the  vertex.     The  intersection  of  this 
perpendicular  with  the  axis  of  .-»,  will  be  the  focus 

287.  Sen.  2.  —  Having  given  the  axis  and  focus,  to  draw  a  tangent  and  a 
normal  at  P,  take  FT  =  FP  =  FE  and  draw  PT  and  PE. 

288.  Sen.  3.—  Having  the  axis  and  focus,  a  tangent  may  be  drawn  making 


184 


PROPERTIES   OF   PLANE   LOCI. 


any  given  angle  with  the  axis  of  .r,  by  making  PFX  =  twice  the  given  angle, 
and  drawing  a  tangent  from  the  point  where  P  F  intersects  the  curve. 


SECTION  XV, 
Special  Properties  of  the  Conic  Sections, 

[NOTE. — The  importance  of  the  Conic  Sections  renders  it  necessary  that  their  properties  should 
be  more  fully  developed  than  is  found  expedient  in  a  compendious  presentation  of  the  subject  of 
the  General  Geometry,  and  hence  this  section.  Similar  sections  might  be  added  on  other  curves, 
as  of  the  cycloid,  the  catenary  ;  or  sections  discussing  the  loci  embraced  by  equations  of  the  3rd 
degree,  or  the  4th  degree,  etc.  But  these  subjects  are  not  of  sufficient  importance  to  require 
treatment  in  an  elementary  course,  nor  capable  of  being  epitomized  so  as  to  be  brought  within 
proper  limits  for  such  a  course.  Those  who  wish  to  pursue  the  subject  farther  will  find  Salmon's 
Conic  Sections  and  Higher  Plane  Curves  in  two  volumes,  or  Price's  Infinitesimal  Calculus  in 
four  large  volumes,  the  best  English  resources.  Todhunter's  four  volumes,  two  on  the  Calculus, 
and  two  on  the  Co-ordinate  Geometry,  are  also  among  the  most  valuable  recent  treatises.  The 
author  of  this  volume  proposes  to  prepare  a  second  volume  on  loci  in  space,  and  a  more  extended 
course  in  the  Calculus,  as  soon  as  he  is  able.] 


(a)  KADII  VECTORES  AND  THE  ANGLES  WHICH  THEY  MAKE  WITH 

A  TANGENT. 

289.  BET. — A  Radius    Vector  of  a  conic   section  is  a  line 
drawn  from  a  focus  to  a  point  in  the  curve. 

290.  Prop. — In  an  ellipse  the  sum  of  the  radii  vectores  to  any 
point  in  the  curve  is  constant  and  equal  to  the  transverse  axis,  and  in  the 
hyperbola  the  difference  is  constant  and  equal  to  the  transverse  axis. 

DEM.     P  being  any  point  in  the  curve, 
let  PF  =  r  and  PF'  =  r'.     We  have 


also  r'=  V  p  D2+  D  F'2=  v/^-f  (Ae+x)*.        Q 
But  1/2  =  (A*  _  tf^i  _  C2)  (52}.     Sub- 
Btituting    this    value    of    y2,    we    have 
r  =  V(A*  —  &)(!  —  e«)  -f  (Ae  —  x)*  = 


—  2Aex  4-  e?&  =  A  —  ex  ;    and 


Fio.  195. 


r'  =  V(A*  —  x*)(l  —  e*)  +  (Ae  -f-  «)*  =  vOi*  +  VAex  +  e*x*  =  A  -f  ex.     Adding, 
r'  +  r  =  2A,  for  the  ellipse. 

In  the  hyperbola  A  —  ex,  is  negative,  since  ex  >  A,  and  we  write  r'  =  A  -f-  ex, 
and  r  =  ex  —  A.     Whence,  subtracting,  r'  —  r  =  2A.     Q.  E.  D. 

291.  COB.  —  The  length  of  a  radius  vector  drawn  to  the  nearer  focus 
is  r  =  A  —  ex,  and  to  the  more  remote  r'  =  A  -f  ex. 


SPECIAL  PEOPEBTIES    OF   THE   CONIC    SECTIONS. 


185 


292.  SCH. — The  principles  enunciated  in  this  proposition  afford  very 
simple  means  for  constructing  the  loci  mechanically.  For  the  ellipse  takei 
a  string  equal  in  length  to  the  transverse  axis,  and  fastening  its  ends  at  the 
foci,  put  a  pencil  against  the  string  and  move  it  around  the  perimeter  of 
the  curve,  keeping  the  string  tense. 
Thus  F'PF  Fig.  195,  represents  the 
string,  and  P  the  pencil  when  the 
point  P  is  located. 

To  construct  an  hyperbola,  take  a 
ruler  AB,  and  a  string  BPF  ;  mak 
ing  the  string  shorter  than  the  ruler 
by  the  length  of  the  transverse  axis 
of  the  required  hyperbola ;  fasten 
one  end  of  the  string  to  one  end  of 
the  ruler,  as  at  B,  fasten  the  other  end  of  the  string  at  one  focus,  as  at  F,  and 
the  other  end  of  the  ruler  at  the  other  focus,  as  at  F'.  Place  a  pencil 
against  the  string  and  bear  it  against  the  side  of  the  ruler,  as  at  P,  and 
keeping  the  string  tense  move  the  pencil  around  the  curve.  It  is  evident 
that  F'P  —  PF  =  ZA  in  all  positions  of  P.  Hence  P  traces  the  curve. 
To  trace  the  other  branch  the  attachments  have  to  be  changed,  so  that 
the  free  end  of  the  string  shall  be  attached  at  F',  and  the  end  of  the 
ruler  at  F. 


FIG.  196. 


203.  Prop* — The  radii  vectores  drawn  to  any  point  in  an  ellipse  of 
hyperbola  make  equal  angles  with  the  tangent  at  that  point. 

DEM.— Let  PF  and  PF',  be  the  radii 
vectores,  and  M  T  the  tangent,  Fig's.  195, 
197.  ThenFPT^F  PM.  For,AT  = 

—  (137,  Ex-  1,  or  270,  Ex.  4),  and  A  F 

x 

=^A  F  =  Ae.  Hence,  in  the  ellipse,  FT= 
-t(A  —  ex),  and  F'T  =  -(A  +  ex)  ; 

and  in  the  hyperbola  FT"  =—(ex  — A), 
and  F'T  =  -(ex  -f-  A).     Wherefore  in 


M 


Fia.  197. 


either  case  we  have  FT  :  F'T  : :  r  :  r'  (291).  Now  drawing  F'M  parallel  to 
PF  we  have  FT  :  F  T  ::  PF  :  F  M,  or  r  :  r'  ::r  :  F'M.  .• .  F'M=r'  = 
F'P,  and  F'MP==  FPT=  F'PM.  Q.  B.  D. 

204.  COB. — In  the  ellipse  the  normal  bisects  the  angle  included  by  the 
radii  vectores  to  the  same  point ;  and  in  the  hyperbola  it  bisects  the  angle 
included  by  one  radius  vector  and  the  other  produced. 


186 


PROPERTIES   OF   PLANE   LOCI. 


295.  SCH.— This  principle  affords  one  of  the  most  convenient  methods 
of  drawing  tangents  to  these  curves. 

1st.  To  draw  a  tangent  th?*ough  a  given  point  in  the  curve.  Let  P,  Fig's. 
195,  197,  be  the  point.  Draw  the  radii  vectores  to  the  point  and  bisect  the 
included  angle  for  the  hyperbola,  or  the  angle  included  by  one  radius  vector 
and  the  other  produced  in  the  case  of  the  ellipse. 

2nd.    To  draw  a  tangent  from  a  point  with 
out  the  curve.     Let  P  be  the  point.     Join 
P  with  the  nearer  focus,  and  from  P  as  a 
centre  pass  an  arc  of  a  circle  through  that 
focus.     From  the  other  focus,  with  a  radius 
equal  to  the  transverse  axis,  strike  an  arc 
cutting  the  former  as  at   D  and  D'.     Join 
D  and  D'  with  F',  and  T  and  T'  are  the 
points  of  tangency.     To  prove  this  for  T, 
join  D  and  F,  and  F  and  T.     Now  F'T-f- 
TF  =  F'D,  since  each  =  2A     Hence 
TD  =  TF.      Moreover    DP  =  PF. 
Hence  "TP  is  perpendicular  to  DF,  and 
angle  DTP:=MTF=PTF.  Whence 
we  know  that  PM  is  tangent  at  P.    In 
a  similar  manner  PM'  can  be  shown 
perpendicular  to  FD',  and  hence  tan 
gent  at  T'.      [The  student  should  com 
plete  the  figure  and  give  the  demon 
stration  in  the  case  of  the  hyperbola.]  FIG.  199. 

3rd.  If  a  circle  be  described  on  PF  and  another  on  CB,  the  lines  passing 
from  P  through  their  intersections  are  tangents  to  the  curve,  and  this 
whether  P  is  in  or  without  the  curve.  [Why  ?  The  student  should  be 
able  to  answer  after  having  read  in  this  section  through  the  subject  of  con 
jugate  diameters.]  This  method,  however,  is  impracticable  when  P  is  with 
out  the  curve,  as  it  does  not  indicate  the  precise  point  of  tangency. 


206.  Prop.  —  In  the  parabola  the  radius  vector  drawn  to  the  point 
of  tangency  makes  the  same  angle  with  the 
tangent   as  a  diameter    through   the  same 
point,  or  as  the  tangent  does  with  the  axis  of 
abscissas. 

DEM.—  Let  PF  be  the  radius  vector,  PF'  the 
diameter,  and  MT  the  tangent.     Then  FPT 
=  FPM.   For,AT=x(14O,or272,  Ex.  1), 
and  A  F  =  ]p.   Hence  FT  =  x  -j-  fa.  But  PF 


FT 

=  FP,  and  angle  FPT=  FTP=F  PM.  Q.E.D. 


FIG.  200. 


SPECIAL  PROPERTIES   OF  THE   CONIC   SECTIONS. 


187 


20 '7 '•  COR. — In  the  parabola  the  normal  bisects  the  angle  included  by 
the  radius  vector  and  a  diameter  at  the  same  point  in  the  curve. 

29S.  Sen. —  To  draw  a  tangent  to  the  point  P  in  the  parabola,  draw  the 
radius  vector  and  the  diameter  to  the  point,  produce  one  of  them  (as  PD) 
and  bisect  the  angle  thus  formed. 

To  draw  a  tangent  from  a  point  without  as  P",  join  the  point  with  the 
focus,  from  P"  as  a  centre,  pass  an  arc  of  a  circle  through  the  focus,  and 
through  its  intersections  with  the  directrix  draw  diameters.  The  vertices 
of  these  diameters  are  the  points  of  tangency  on  the  curve,  as  P  and  P'. 
To  prove  this,  observe  that  as  PF  =  PD,  and  P"F  —  P"D,  P"P  is  per 
pendicular  to  DF  and  angle  FPP"  —  DPP"  =  MPF.  [Let  the  stu 
dent  give  the  proof  for  the  point  P'.] 

[NOTE.— The  properties  demonstrated  in  these  propositions  give  elliptic,  hyperbolic,  and 
parabolic  reflectors  their  peculiar  properties.  Thus,  rays  of  light,  sound,  or  heat  diverging  from 
one  focus  of  an  elliptic  reflector  are  converged  at  the  other  ;  diverging  from  one  focus  of  an 
hyperbolic  reflector,  they  diverge  after  reflection  as  though  they  proceeded  directly  from  the 
other  focus  ;  and  diverging  from  the  focus  of  a  parabolic  reflector,  they  are  reflected  parallel. 
Conversely  to  the  last,  incident  rays  parallel  to  the  axis  are  concentrated  at  the  focus  of  a  para 
bolic  reflector.] 


299.  Prop. — The   rectangle  of  the  perpendiculars  from  the  foci 
upon  the  tangent  of  the  ellipse  or  hyperbola  is  constant,  and  equal  to  the 
square  of  the  semi-conjugate  axis. 

DEM. — Let  L'T  be  a  tangent  at  any  point  P,  and 
FL  a  perpendicular  from  the  focus  upon  it.     Pro 
duce  FL  till  it  meets  F'P,  produced  if  necessary, 
in  D,  and  draw  AL.     Since  AF  =  AF'  and   FL 
=  LD,  AL  is  parallel  to  F'D  and  equal  to  £F  D 
=  A  B.     Hence  the  foot  of  a  perpendicular  from  ihe 
focus  upon  a  tangent  lies  in  the  circumference  of  a  circle 
described  on  the  transverse  axis.     Now  let  F'L'  be 
the  perpendicular  from  the  second  focus  upon  the 
tangent  L  T,  we  are  to  show  that   FL  X  F'L'  = 
B*.     Join   A  and   L'  and  produce  the  line 
till  it  meets    LF    produced  in   L".     Then     . 
the  triangles  AL'F'  and  AL"F  are  equal, 
and   AL"  =  AL',  and  L"  is  in  the  cir 
cumference  of  the  circle  described  on  C  B. 
Finally,    FL  X    F'L'  =  FL  X    FL"  = 
FB   X    FC  (CB  and  LL"  being  chords 
in  the   same   circle).      But    FB  X   FC  — 
(A  +  Ae)(A  -  Ae)  =  A* (I  -  c=)  =  &  (49, 
3rd  and  7th).        .-.   FL   X    F'L'   =  B*.  FlG>  202. 

Q.  E    D. 

300.  COR. —  The  semi-conjugate  axis  is  a  mean  proportional  between 
the  focal  distances. 


188 


PROPERTIES   OF  PLANE   LOCI. 


(6)  SUPPLEMENTARY  CHORDS  AND  CONJUGATE  DIAMETERS. 

30  1.  DEF.  —  The  term  Ordinate,aa  used  in  connection  with  the 
conic  sections,  may  mean  any  line  drawn  from  a  point  in  the  curve  to 
any  diameter,  and  parallel  to  a  tangent  at  the  extremity  of  that 
diameter. 

302.  DEF.  —  Supplementary    Chords    are   chords   drawn 
from  any  point  in  the  curve  to  the  extremities  of  any  diameter. 

303.  DEF.  —  One  diameter  is  said  to  be  conjugate  to  another 
when  it  is  parallel  to  a  tangent  at  the  extremity  of  the  latter. 


304.  Prop.  —  In  an  ellipse  the  rectangle  of  the  tangents  of  the  angles 
which  a  pair  of  supplementary  chords  make  with  the  transverse  axis  is 

equal  to  --  . 

DEM.  —  Let  PC  and  PB  be  supplemen 
tary  chords  drawn  to  the  axis.  Let  the 
angles  PCX  and  PBX  be  represented  by 
ct  and  «',  and  tan  ex.  =  a,  and  tan  a.'  =  a'. 

Then  aa'  =  --  —  ,  A  and  B  being  the  semi- 

^1* 

axes.  The  equation  of  PC  is  y  =  a(x+  A), 
and  of  PB  y  =  a'(x  —  A),  disregarding 
the  signs  of  a  and  a',  since  PC  and  PB 
are,  as  yet,  any  lines  passing  through  C,  and  B.  For  the  intersection  of  thest. 
lines  these  equations  are  simultaneous,  and  we  may  have  y-  =  aa'(xz  —  ./I2). 
Again,  when  the  point  of  intersection  is  in  the  ellipse,  this  result  is  simultaneous 

7?2 

with  the  equation  of  the  curve,  A-y*  -{-  B-x-  =  A*B*,  or  y*=  —  "nC**  —  ^  • 

•n  -2 

whence  combining  the  two,  we  have  oa'=  —  —  .     Q.  E.  D. 


.  COR.  1.  —  By  a  similar  course 
of  reasoning,  or  by  simply  changing 
the  sign  of  B",  we  have  for  the  hyper- 

B2 

bola  aa'  =  —  . 

A2 

306.    COR.   2.—  In  the    ellipse,  if 
supplementary  chords  are  drawn  to  the     ,-p, 
extremities  of  the  conjugate  axis  aa'=  FIG.  204. 

B« 

—  —  (the  same  as  before)  if  the  angles  are  measured  from  the  axis  of  x, 

A." 

A2 

but  aa'  =  —  —  if  they  are  measured  from  the  axis  of  y. 


SPECIAL  PROPERTIES  OF  THE  CONIC   SECTIONS.  189 

DEM.— The  equations  of  P'E  and  P'  D,  Fig.  203,  are  respectively  y  —  B  —  ax 
and  y  -f-  B  —  a'x  ;  whence  y*  —  B*  =  aa'x2.  From  A3y*  -f-  J5  a;2  —  A2B*  we  have 

B*  J3* 

--*2.     .-.  «a=    --. 

1  1  A1 

of  y  we  observe  that  for  a  we  shall  have ,  and  for  —  a',  — .     .  •     aa'  = 

a  a  B* 

307.  COR.  3. — In  the  hyperbola,  if  supplementary  chords  are  drawn 
from  any  point  in  the  conjugate  curve  to  the  extremities  of  the  conjugate 

B2 

axis  aa'  =  —  if  the  angles  are  reckoned  from  the  axis  of  x,  but  aa'  = 

—  if  they  are  reckoned  from  the  axis  of  j. 

DEM. — [Let  the  student  supply  the  demonstration,  and  also  show  that  if  the 
chords  be  drawn  from  a  point  in  the  x  hyperbola  to  the  extremities  of  the  conju 
gate  axis,  aa'  is  not  constant.  ] 

308.  COR.  4. — In  the  circle   this  relation  becomes  aa'  =  —  1,  or 
1  -{-  aa'  =  0  ;   which   shows  that  the  chords*  are  perpendicular  to  each 
other,  which  is  a  well  known  property  of  the  circle.     In  the  equilateral  hy 
perbola  the  relation  is  aa'  =  1,  or  a  =  '— ,  signifying  that  the  angles  are 
complementary. 

300.  COR.  5. — If  one  of  two  supplementary  chords  to  either  axis  is 
parallel  to  one  of  two  drawn  to  the  other  axis  the  other  two  are  parallel. 

DEM.— In  either  Fig.  203  or  204=  if  P'E  is  parallel  to  PC,  P'D  is  parallel  to 

PB.     For  we  have  in  case  of  each  set  of  chords  aa'  =  =F  -p     •'  •  If  a  is  the 
same  in  each,  a'  is  also. 

31O.  Sen. — The  —  sign  in  the  formula 

aa'  = indicates  that  a  and  a'  have  op- 


posite  signs  in  the  ellipse.     Thus  P  being     \\ 

the  point  from  which  the  chords  are  drawn,       **""" 

PBX  >  90°  and  <  180°  gives  —  a',  PCX 

<  90°,  gives  -f-  a.     Again  P"BX  being  an  FIG.  205. 

angle  between  180°  and  270°,  tan  P"BX  =  +  a,  but  P"CX  being  between 

270°  and  360°,  its  tangent  is  —  a.     In  a  similar  manner  the  -f  sign  in  the 

B* 

formula  aa  =  -\ signifies  that  a  and  a  have  always  the  same  sign,  as 

A* 

may  be  readily  observed  from  a  figure. 


311.  J?rob. — To  discuss  the  angle  included  between  supplementary 

chords  to  the  transverse  axis  of  an  ellipse. 


190  PROPERTIES  OF  PLANE  LOCI. 

SOLUTION.— Let  V  be  the  included  angle  CPB  ;  then  tan  V  =  1  +aa~  = 

Limiting  the  discussion  to  the  upper  segment  CPB,  Fig.  205,  a'  is 

always .     Hence  tan  V  is  — ,  and  V  is  always  an  obtuse  angle. 

*Again,  as  V  is  a  variable  angle,  we  may  inquire  when  it  is  a  maximum.     Differ 
entiating  a'  +  ~-  with  respect  to  a,  and  putting  the  result  =  0,  we  have1 

a'  =  -±-  _.     The  ambiguous  sign  is  explained  by  the  fact  that  the  result  applies 
equally  well  to  either  angle  PBX,  or   PCX.     Hence  V  is  a  maximum  when 

tan  PBX— ,  or  tan  PCX  =  -r,  i.  e.,  when  P  is  at  D. 

A  A 

To  find  the  value  of  V  when  it  is  a  maximum,  we  have  simply  to  substitute 
—  -  for  a  in  tan  V  ;  whence  tan  V  =  —  ^ ^ 

Sen.— The  angle  included  by  supplementary  chords  to  the  conjugate 
axis  of  an  ellipse,  and  also  the  corresponding  cases  in  the  hyperbola,  may 
be  discussed  in  a  similar  manner ;  but  the  results  are  not  important. 


3  12.  Prop.  —  If  one  of  two  supplementary  chords  to  the  transverse 
axis  of  an  ellipse  is  parallel  to  a  tangent,  the  other  is  parallel  to  the  diam 
eter  drawn  through  the  point  of  tangency,  and  conversely. 

DEM.  —Let  P  B  be  parallel  to  M  T,  then  is 
PC  parallel  to  DD'.  Let  tan  PCX  =  a, 
tan  PBX  =  a',  tan  DAX  =  an  and 
tan  MTX  =  a/.  Now  the  equation  of  D  D' 

is  y  =  «jX  ;  whence  #j  =  -.     Also  tan  MTX 


=  _  —(136,Ec.l,  or  £0f».Hence,  «,«!'= 
A*y 

But  oo'  =  —  3p-     •••  aa>  =  «i«i  '»  and  if  a'  =  FlG<  2°6' 

a/,  a  =  flj.     Conversely,  if  a  =  alt  a'  =  a/.     Q.  E.  D. 

313.  COR.  1.  —  The  same  property  exists  in  the  hyperbola  and  is  dem 
onstrated  in  the  same  way. 

314.  Sen.  —  This  property  affords  a  convenient  method  of  drawing  tan 
gents.     Thus  to  draw  a  tangent  at  D  Fig's.  206,  207,  draw  the  diameter  D  D', 
the  chord  PC  parallel  to  it,  the  supplementary  chord  PB,  and  through  D 
draw  MT  parallel  to  PB. 

*  Students  wlio  have  not  read  the  Calculus  will  omit  this  paragraph. 


SPECIAL  PROPERTIES   OF  THE   CONIC   SECTIONS. 


191 


To  draw  a  tangent  parallel  to  a  given 
line,  or  what  is  the  same  thing,  making 
a  given  angle  with  the  axis  of  x,  draw 
a  chord  P  B  parallel  to  the  given  line 
EF,  or  making  the  given  angle  PBX, 
draw  the  supplementary  chord  PC, 
and  the  diameter  D  D'  parallel  to  PC. 
Through  D  draw  MT  parallel  to  PB 
and  it  is  the  tangent  sought. 


FIG.  207. 


315.  COR.  2. — If  one  of  two  supplementary  chords  to  the  transverse 
axis  is  parallel  to  a  diameter,  the  other  chord  is  parallel  to  the  conjugate 
diameter,  for  the  latter  diameter  is  parallel  to  a  tangent  at  the  vertex  cf  the 
former  (303).  Hence,  also,  if  one  diameter  is  conjugate  to  another, 
reciprocally,  the  latter  is  conjugate  to  the  former. 

Ex.  1.  In  an  ellipse  whose  axes  are  8  and  G,  one  supplementary 
chord  to  the  transverse  axis  makes  an  angle  with  that  axis  whose 
tangent  is  2  ;  what  angle  does  the  other  make  ? 

;,  SOLUTIONS. — Arithmetically.  Using 


the  formula    aa' 
.B  —  3,  and  A  = 


2, 
whence  a'  = 


— ,  a 


or  the    angle  is    164°  18' 
Geometrically.    Construct  C 


32' 

nearly. 

the  ellipse  with  C  B  =  8  and  D  E 
=  6.  Make  PCB  =  tan-' 2  and 
draw  P  B.  The  angle  P  B I  is  the 

one  required,  whose  tangent  is  found 

g 

by  measurement  to  be  about . 

32 


FIG.  208. 


Ex.  2.  The  same  as  above,  the  tangent  of  the  angle  being 
the  axes  10  and  6. 


5,  and 


Ex.  3.  The  same  numbers  as  in  Ex.  1,  applied  to  the  hyperbola. 
What  are  the  co-ordinates  of  the  point  in  the  curve  from  which  the 
chords  are  drawn? 

J?2  9 

SUG. — For  the  solution  of  the  last  question  we  have  aa'  =  —    or  a  =  — ,  «' 


— (x  -J-  4)  to  find  x  and  y,  which  are  nearly  5. 3 
32 


being  2  ;  y  =  2(x  —  4)   and  y 
and  2.G. 

Ex.  4.  In  an  ellipse  whose  axes  are  8  and  6  find  the  angle  included 
by  the  supplementary  chords  to  the  transverse  axis,  from  the  point 
x  =  1. 


19(2 


PROPERTIES   OF   PLANE   LOCI. 


310.  I*vol). —  To  dram,  geometrically,  a  pair  of  supplementary 
chords  to  the  transverse  axis  of  a  given  ellipse  so  that  the  chords  shall 
include  a  given  angle. 

SOLUTION.  —  Upon  the 
transverse  axis  describe 
a  segment  of  a  circle 
B  P  M  A  which  shall  con 
tain  the  given  angle — in 
this  case  ABC.  From 
the  intersection  of  this 
circumference  with  the 
ellipse,  as  from  P  or 
P',  draw  supplementary 
chords.  The  pupil  may  FIG.  209. 

give  the  reason,  and  also  show  how  the  construction  will  indicate  the  impossibility 
in  case  the  angle  given  is  larger  or  smaller  than  can  be  included  by  chords  in  the 
given  ellipse. 

317*  GENEKAL  SCHOLIUM. — From  the  preceding  articles  in  this  section 
it  is  evident  : 

1st.  That  to  draw  a  diameter  conjugate  to  a  given  one,  we  may  draw  a 
tangent  through  the  extremity  of  the  given  diameter  and  then  draw  a  di 
ameter  parallel  to  this  tangent ;  or  we  may  draw  one  of  two  supplementary 
chords  parallel  to  the  given  diameter,  and  drawing  the  other  supplementary 
chord,  draw  the  second  diameter  parallel  to  the  last  chord. 

2nd.  To  draw  a  pair  of  conjugate  diameters  which  shall  include  a  given 
angle,  draw  a  pair  of  supplementary  chords  which  shall  include  the  angle, 
and  parallel  to  these  draw  a  pair  of  diameters. 

3rd.  If  a  be  the  tangent  of  the  angle  which  one  diameter  makes  with  the 
transverse  axis  and  a'  the  tangent  of  the  angle  which  the  other  makes, 

B*  B* 

aa  =  .»-  — -  in  the  ellipse,  and  aa'  =  —  in  the  hyperbola.    Letting  tan""1  a 


and  tan-1  a'  be  respectively  a  and  a',  and  we  have  a 


sin  a 


and  OL 


sin  a' 


cos  a  cos  a  • 

Substituting  these  values,  there  results  in  the  case  of  the  ellipse  A2  sin  a  sin  a' 
-f-  B-  cos  a  cos  a'  =  0,  and  of  the  hyperbola  A*  sin  a  sin  a'  —  B*  cos  a  cos  a' 
=  0,  formula*  which  are  sometimes  referred  to  as  the  equations  of  condition 
of  conjugate  diameters. 


4th.  From  the  relation 


—  we  may  also  solve  the  2nd   above. 


Thus,  if  ft  be  the  given  included  angle,  ft  =  a  —  a,  and  tan  /3  = 


1-Hoa 


A*a' 


j-  ;  whence  a'  can  be  determined,  as  all  the  other  quantities  are 


supposed  known.. 


SPECIAL  PROPERTIES  OF  THE  CONIC  SECTIONS.       193 

318.  Prob.  —  To  investigate  the  relation  between  conjugate  diameters 
and  the  axes. 

DEM.  —  The  equation  of  the  ellipse   and   hyperbola  referred  to  the  conjugate 
diameters  2  At  and  22?2  is  Aftf  ±  B^x*  =  ±  A^B^  (127,  Ex>s.  10  and  11)  ;  the 
-f-  sign  applying  to  the  ellipse  and  the  —  sign  to  the  hyperbola.  Transforming  this 
equation  so  that  the  reference  shall  be  to  the  axes,  by  means  of  tbeformulce 
y  cos  a  —  x  sin  a  x  sin  a  —  y  cos  a' 


»  ,  ScH.), 

—  a)  sm  (a  —  a) 


we  have 

—  2^1  2  5a-y  cos  a  sin  a  -f- 


_  2  .         , 

db  2W-  cos*  a  =f  2B,*xy  cos  a'siu  a'  ±  '     ~ 


Comparing  this  with  A°y*  ±  B?x'2  =  ±  A^B*,  we  have 
(1) 
(2) 

(3)  ^I2acos  a:  sin  a  ±  _B2  2  cos  a  sin  a'  =  0,  and 

(4)  A.^B^Bm^(a  —  a)  =  A*B*. 

1.  Adding  (1)  and  (2)  and  remembering  that  sin2  -f-  cos2  =  1,  we  have 
-<V  db  B^  =  ^l2  ±  B2,  that  is 

(a)  In  Me  ellipse  the  sum  of  the  squares  of  conjugate  diameters  is  constant,  and 
equal  to  the  sum  of  the  squares  on  the  axes.  In  the  hyperbola  the  difference  of  the 
squares  is  constant  and  equal  to  the  difference  of  the  squares  on  the  axes. 

2.  Making  Aoy=B^  (3)  gives  —  -  -  -  -  r  =  =F  1  ;  but  as  A2  and  J?2  are  conju- 

cos  a  sin  a 

sin  a  sin  a  B*       ,,  .,  .   .   .        ,.  „   ,, 

gate   tan  a  tan  a    =  --  =  =F   —  .      Multiplying  the  members  of  these 
cos  a  COH  a  A* 

sin*  a         B*          sin  a  11 

equations  and  reiecting  common  factors  we  have  -  =  —  -,  or  -  =  ±  -, 

cos^a         A*         cos  a  A 


the  —  sign  characterizing  the  ellipse  since  a  is  obtuse  in  the  ellipse,  and  the  -f- 
sign  characterizing  the  hyperbola,  as  in  it  a.  and  a  are 

A  7? 

acute.     Hence  --  =  -:  -  indicates  the  position 
cos  a          sin  a 

of  equal  conjugate  diameters  in  the  ellipse,  and  --  ; 

T> 

=  -  in  the  hyperbola.     From  Fig.  210  we  see  that 
sin  a 

A  D  and  A  F  meet  this  condition  in  the  ellipse  ;   for 


B 

Hence 


sm  a 

(b)  In  the  ellipse  the  conjugate  diameters  which  fall  upon  the  diagonals  of  the  rect 
angle  on  the  axes  are  equal,  and  a  and  a'  are  supplementary. 

3.  In  the  hyperbola  in  general,  the  condition ;  = is  met  only  when  the 

two  diameters,  as  G  F  and  DE  Fig.  212,  coincide  and  fall  on  the  asymptote. 
Hence,  in  the  hyperbola,  asymptotes  are  the  analogues  of  the  equal  conjugate 
diameters  of  the  ellipse.  But  from  A^  —  B2*  =  A*  —  JB*,  we  observe  that  if 
A  ass  B,  A 2  =  B2,  independently  of  a  and  a'.  Hence 


194 


PKOPEKTIES   OF   PLANE  LOCI. 


(c)  In  the  hyperbola,  in  general,  there  are  no  equal  (finite)  conjugate  diameters;  but, 
in  the  equilateral  hyperbola,  any  pair  of  conjugate  diameters  are  equal  each  to  each. 
The  conjugate  diameters  of  the  equilateral  hyperbola  find  their  analogues  in  the 
diameters  of  a  circle. 

4.  Making  A2  =  B2  in  A.^  -f  B2*  =  A*  -j-  B*,  we  find  that 

(d)  The  length  of  one  of  the  equal  conjugate  diameters  of  an  ellipse  is  s/2  v/A^-f  B*. 
.-.     The   semi-conjugate    diameter:     the    semi-diagonal  on  the  axes  ::  1  :  \/2. 

5.  Extracting  the  square  root  of  both  members 
of  (4),  we  have  A2B2  sin  (a'  —  a)  =  AB  ;  which 
signifies  that 

(e)  The  parallelogram  formed  by  tangents  drawn 
through  the  vertices  of  any  pair  of  conjugate  dia 
meters  is  constant,  and  equal  to  the  rectangle  on  the 
axes.     This  will  be  more  apparent  from  Fig's.  211, 
212.    D  A  F  =  («'  —  a),  and  D  A  =  Bt ;  whence 
AO  =  -Z?2  sm  (<*' —  <*)•    Hence  A2B2  sin  (a'  —  a) 
=  area   AFHD  =  4LKIH  =  AB  =  J  the 

rectangle  on  the  axes.     .  • .    L  K I  H  =  the  rectangle  on  the  axes. 

Ex.  1.  Write  the  equation  of  an  el 
lipse  referred  to  a  pair  of  conjugate 
diameters  whose  lengths  are  8  and  6, 
and  the  included  angle  tan"1  ( —  2). 
Having  written  the  equation  con 
struct  it  as  in  Chapter  I.,  Sec.  II. 

Ex.  2.  Write  the  equation  of  an 
hyperbola  referred  to  conjugate  dia-  FIG.  212. 

meters  whose   lengths  are  12  and  8,  and  whose   included  angle  is 
tan-12.     Construct  as  in  the  last  example. 

Ex.  3.  In  an  ellipse  whose  axes  are  8  and  6  what  is  the  length  of  a 
diameter  which  makes  an  angle  of  45°  with  the  axis  of  #?  What  is 
the  length  of  its  conjugate  ? 

7?2 

SUG'S. — From  the  relation  aa'  =  —  — ,  we  learn  th»t  the  conjugate  diameter  makes 

with  the  axis  of  x  an  angle  of  150°  39'  nearly.     Hence  A2B2  sin  (a'  —  a)  =  AB, 

12 

becomes  A2B2  sin  105°  39'  =  12,  or  A2B2  =  -^^—.    Also^22  +  Bg*  =  A*  +  -B2 

,9b29o 

gives  A^-}- B.2*  =  25.     These  two  equations  will  give  the  values  of  A2  and  B2. 

Ex.  4.  In  an  ellipse  whose  axes  are  8  and  6,  what  are  the  sides  of 
the  circumscribed  parallelogram  whose  sides  are  parallel  to  the  equal 
conjugate  diameters  ?  What  is  the  altitude  of  this  parallelogram  ? 

Altitude,  6.79  nearly. 


SPECIAL  PROPERTIES  OF  THE  CONIC  SECTIONS. 


195 


(c)    PKOPEKTIES  OF  OKDINATES. 

310.  JProp. — The  squares  of  ordinates  to  the  transverse  axis  of  an 
ffllipse  are  to  each  other  as  the  rectangles  of  the  segments  .into  which  they 
Respectively  divide  the  axis. 

DEM.— Let  PD  =  y,  P'D'  =  i/'.  AD  = 
x,  A  D'  =  x',  then  A^  -f  B*x  =  A*B2,  and 
A"y'2  -f~  B'*x'2  =  A*B'* ;  whence  y2  — 

-7-  (A1  —  x2)  and  y'2  =  -— (A2  —  x'2).     Divid- 
Ai  A2 

ing  and  rejecting  the    common  factor  we 

_*'}(A- 
A*  — 


-;or 


or    ::    CD 

Q.  E.  D. 


X 


(A  -f-  x)(A  —  x) 
-x)  :(A+*)(A  —  x') 
DB    :    CD     X    D  B. 


E 

FIG.  213. 

320.  COR.  1. — The  square  of  any  ordinate  to  the  transverse  axis  of 
an  ellipse  is  to  the  rectangle  of  the  segments  into  which  it  divides  that  axis, 
as  the  square  of  the  conjugate  axis  is  to  the  square  of  the  transverse. 


DEM. — In  the  above  proportion  if  y' 
have  y2  :  &  : :  C  D  X  D  B  :  A>.     .  • . 


=  Q  A  =B,  {A  +  x')(A  —  x')  =  A\  and  we 
f2  :  C  D  X  D  B  : :  ±B*  :  4^42.     Q.  B.  D. 


321.  COR.  2. — The  latus  rectum  is  a  third  proportional  to  the  trans 
verse  and  conjugate  axes. 

DEM. — In  the  last  proportion  let  y  become  the  focal  ordinate  P"F,  which  call  p, 
and  CD  X  DB  becomes  CF  X  FB=(^l-f-  c)(A  —  c),  c  being  AF.  Now 
(A  -f  c')(A  —  c)  =  A*  —  c2  =  B*>  hence  p*  :  B*  : :  B2  :  A*,  or  2^4  :  2J5  : :  25  :  2p. 

Q.  E.  D. 

322.  Sen.— The  prop 
erties  demonstrated  in 
this  proposition,  and  in 
the  1st  and  2nd  corolla 
ries,  are  equally  true  for 
the  hyperbola,  and  can 
be  proved  in  the  same 
way.  In  the  case  of  the 
hyperbola,  however,  the 
statement  should  be, 
The  rectangles  of  the  dis 
tances  from  the  feet  of 
the  ordinates  to  the  ver 
tices,  instead  of  "the 
rectangles  of  the  segments,  etc.,"  as,  in  this  case  the  ordinates  do  not 


196 


PROPERTIES   OF  PLANE  LOCI. 


divide  the  axis,  but  fall  upon  its  prolongation ;   so  that,  in  Fig.  214,  we 
have  ys  :  y'2  : :  CD  X  BD  :  CD'  X  BD'. 

323.  COR.  3. — In  the  case  of  the  circle  COR.  1st  shows  that  the  square 
of  the  ordinate  equals  the  rectangle  of  the  segments  into  which  it  divides 
the  diameter — a  well  known  property. 

324.  COR.  4. — This  proposition  and   COR.   1st  may  be  asserted  of 
ordinates  to  the  conjugate  axis.     [Let  the  student  give  the  proof  and  a 
figure  to  illustrate  it.] 


25.  COR.  5. — This  proposition  and  COR.  1st  may  also  be  asserted  of 
ordi  nates  to  ANY  diameter  of  an  ellipse  or 
an  hyperbola. 

DEM. — The  corollary  can  be  proved  in  the 
same  way  as  the  proposition,  by  using  the 
equation  of  the  curves  referred  to  conjugate 
diameters  (127,  Ex's.  10  and  11),  since  these 
equations  are  of  the  same  form  as  those  used 
above.  In  the  annexed  figures,  therefore, 
CD 


PD 


the 
-i. 


X    DB    :   CD'    X 


D  B.    Also  PD"  :  CD  X  DB 

326.  COR.  6.  —  From 
the  la~<t  relation,  it  fol 
lows  that  chords  paral 
lel  to  any  diameter  are 
bisected  by  its  conjugate, 


:  A'*. 


FIG.  215. 


=  D'H',  etc.  ;  and 
hence  that  these  curves 
are  symmetrical  with  re 
spect  to  any  diameter. 


327.  Sen.— These  principles,  together 
with  others  already  known,  enable  us  to 
find  the  centre,  axes,  and  foci  of  the  curves, 
geometrically,  when  the  perimeters  alone 
are  given.  Thus,  in  the  case  of  the  ellipse, 
let  the  curve  NHIM  be  given,  to  find  the 
centre,  axes,  and  foci.  Draw  any  two  par 
allel  chords  as  DE  and  BC,  bisect  them 
at  K  and  L,  and  draw  FG  ;  it  will  be  a 
diameter  by  COR.  6.  Bisect  this  diameter 
and  A  will  be  the  centre.  From  A  with 


FIG.  216. 


FIG.  217. 


SPECIAL  PROPERTIES   OF  THE  CONIC  SECTIONS. 


197 


a  radius  sufficiently 
long  to  cut  the  curve, 
construct  the  circle 
HIMN,  join  two  of  the 
intersections,  as  I  and 
H,  and  perpendicular 
to  this  chord  pass  a  line 
through  the  centre ;  it 
will  be  the  axis.  [The 
student  can  readily  fin 
ish  the  problem.] 

The  construction  is 
the  same  for  the  hy 
perbola  except  in  find 
ing  the  conjugate  axis 
when  the  conjugate 
hyperbola  is  not  given.  FIG.  218. 

For  this  purpose  use  the  proposition  in  Con.  1.     In  the  figure,  take 

SR  X  VR  :  RR2  :  :  AV3  :  AO2  ;   whence  AO  can  be  constructed. 


328.  Prop. — In  different  ellipses  upon  the  same  transverse  axis,  the 
corresponding  ordinates  to  the  transverse  axis  are  to  each  other  as  the  con 
jugate  axes  of  the  respective  curves. 

DEM.— We  have  PG2:  CQ  X 
Q  B  : :  AD2 :  AB2,  alsoP^Q2 : 
CQ  X  G  B  : :  A~DT2:  A~B2  and 
P^G2  :  CQ  X  G  B  :  :  AD7'2 : 
A~B2.  .-  .  PG  :  P'G  :  P"Q  :  : 
AD  :  AD' :  AD",  etc.  Q.  E.  D. 

329.  Con.— Any    ordi-    C 
note  to  the  transverse  axis  of 
an  ellipse  is  to  the  correspond 
ing  ordinate  of  the  circum 
scribed  circle  as  the  conjugate 
axis  of  the  ellipse  is  to  the 

transverse. 

^*~. 

FIG.  219. 

DEM.— Let  CD'"B  be  the  circumscribed  circle,  then  as  it  may  be  considered  as 
an    ellipse   with    equal    axes,   we    have  PG    :  P'"G    : :  AD   :  AD'"(=  AB). 

Q.  E.   D. 


198 


PROPERTIES   OF  PLANE  LOCI. 


33O.  Sen. — An  instrument  called  a 
Trammel  is  constructed  upon  the  prin 
ciple  enunciated  in  this  corollary.  It 
consists  of  two  grooved  bars  X '  X ,  Y  Y ', 
fastened  together  at  right  angles,  and 
an  adjustable  arm  PH.  H  and  I  are 
pins  which  can  be  fastened  anywhere 
on  PH,  and  have  heads  on  the  under 
side  which  run  in  the  grooves  of  the 
bars.  Any  point  in  the  movable  bar, 
as  P,  traces  an  ellipse  as  H  and  I  slide 
back  and  forth  in  the  grooves.  To 
prove  that  P  is  a  point  in  an  ellipse  of  FIG.  220. 

which  PH  is  the  semi-transverse  axis  and  PI  the  semi-conjugate,  draw 
AP'"  and  PH  parallel  to  it,  Fig.  219.  Produce  PQ  till  it  meets  HE  drawn 
parallel  to  AB,  in  E.  Then  AP'"  =  A  =  PH.  Again,  P"  G  :  PG  : : 
P'"A  :  PI,  or  ordinate  of  circle  :  ordinate  of  ellipse  :  :  A  :  PI.  And  as  this 
is  true  for  all  positions,  PI  being  made  ==  B,  and  PH  =  A,  P  is  always 
in  the  curve. 

We  may  also  demonstrate  directly  that  the  locus  of  P  is  an  ellipse.  From 
Fig.  220,  using  the  common  notation  PI:PH::PD:PE,  gives  B  :  A  : : 
y  :  VA*  —  &,  or,  squaring,  B*  :  A*  : :  y*  :  A*  —  x* ;  whence  A*y*  -f-  B*x*  = 


Prop. — In  different  ellipses  on  the  same  conjugate  axis,  cor 
responding  ordinates  to  this  axis  are  to  each  other  as  the  transverse  axes 
of  the  respective  curves. 

DEM.— We  have  PG*:DG  X 
G  E  : :  A^B2 :  ACT'  and  RXS2  : 
DQ  X  GE  :  :  A~B'2  :  AD2. 
.-.  PG  :  P  Q  ::  AB  :  AB'. 

Q.  E.  D. 

332.  COR.— Any  ordi 
nate  to  an  ellipse  is  to  the  cor 
responding  ordinate  of  the  in 
scribed  circle,  as  the  transverse 
axis  of  the  ellipse  is  to  the  con 
jugate.  [The  student  may 
make  the  deduction  from 
the  proposition.] 


SPECIAL   PROPERTIES   OF  THE   CONIC   SECTIONS.  199 

333.  Prop. — The  squares  of  ordinates  to  any  diameter  of  a  para 
bola  are  to  each  other  as  their  corresponding  abscissas. 

DEM. — Referred  to  any  diameter,  as  AX  or  AiXt, 

the  equation  of  the  parabola  is  y2  =  2px  (127,  Ex- 12). 
Whence,  letting  y  and  y'  represent  any  two  ordinates, 
as  PD  and  P'D,  or  PJD1  and  P/D/,  and  a;  and 
x'  the  corresponding  abscissas  we  have  y-  =  2pas  and 

Vs        x 
y'2  =  2px'.     Dividing,  —-==:—.     Q.  E.  D. 

334.  COR. — All  chords  drawn  parallel  to  a 

tangent  at  the  extremity  of  a  diameter  of  a  par-  FlG  222 

abola  are  bisected  by  that  diameter. 

335.  SCH. — Having  the  curve  to  find  the  axis  and  focus  of  a  parabola, 
we  draw  any  pair  of  parallel  chords,  and  bisect  them  by  a  right  line.     This 
line  is  a  diameter.     Draw  two  other  parallel  chords  perpendicular  to  the 
diameter  thus  found,  bisect  these  chords  by  a  right  line,  and  it  will  be  the 
axis.     Find  the  focus  by  (164,  or  284). 


(d)   ECCENTRIC  ANGLE. 

336.  DEF.  —  The  Eccentric  Angle  in  an  ellipse  is  the  angle 
formed  with  the  axis  of  abscissas  by  a  line  drawn  from  the  centre  to 
a  point  in  the  circumference  of  the  circumscribed  circle  where  a  pro 
duced  ordinate  meets  it,  that  is  P'"A  B  Fig.  219. 


.  Prop.  —  The  abscissa  of  any  point  in  the  ellipse  equals  the 
semi-transverse  axis  into  the  cosine  of  the  eccentric  angle,  and  the  corres 
ponding  ordinate  equals  the  semi-conjugate  axis  into  the  sine  of  the  same 
angle.     That  is,  letting  cp  represent  the  eccentric  angle, 
x  =  A  cos  cp,  and  y  =  B  sin  cp. 

DEM.  —  In  Fig.  219,  AQ  =  x  =  P'"A  cos  P'"  A  B  =  A  cos  cp.  Also  PQ  =  y 
=  P  I  sin  P  I  Q  =  .Bsin  cp. 

SCH.  —  The  introduction  of  this  angle  is  a  recent  device  to  facilitate  the 
deduction  of  certain  properties  of  the  ellipse.  It  enables  us  to  transform 
an  equation  in  terms  of  rectangular  co-ordinates  (x,  y]  into  one  containing 
but  one  variable,  <p,  which  is  sometimes  of  much  advantage.  We  will  give 
a  few  specimens  of  its  use. 

33  8.  Prop.  —  The  equation  of  a  tangent  to  the  ellipse  in  terms  of 
the  eccentric  angle  is 

A  sin  cp  •  y  -{•  B  cos  cp  -  x  =  AB. 

DEM.  —  The  equation  of  a  tangent  to  an  ellipse  is  A^y'y  -f-  B*x'x  =  A2B*.  As 
(x',  y')  is  a  point  in  the  ellipse,  we  have  x'  =  A  cos  cp,  and  y'  =  B  sin  cp.  Substi 
tuting  these  values  and  dividing  by  AB,  we  have  A  sin  cp>  y  -f-  B  cos  cp  •  x  —  AB. 
Q  E.  D. 


200  PROPERTIES  OF  PLANE  LOCI. 

339.  Prop. —  The  eccentric  angles  of  the  vertices  of  conjugate  diam 
eters  differ  by  90°. 

DEM. — Let  DAB  =  <p,  and  D'AB  =  cp,  be  the  ec 
centric  angles  of  the  vertices  of  the  conjugate  diameters 
PC  and  P'C'.  Letting  (x,  y)  be  P,  and  («,,  j/0  be 
P',  the  equations  of  AP  and  AP'  are,  respectively 

y  =  ax,  or  a  =  -,  and  yL  =  a'xit  or  a'  =  — .     Whence 

B2       T/VI        B  sin  <z>  X  J5  sin  cp'  -r,       non 

aa'  =  —  —  =  —  =  - ^--^—A — ,,  or  tan  cp  tan  <p  FIG.  223. 

A-       xxl       ^4cos</>X  Aco&cp 

=  —  1.     Hence  A  D  and  AD'  are  perpendicular  to  each  other,  and  <p  =  <p  -f- 
90°.     Q.  E.  D. 

3&O.  SCH. — This  proposition  affords  a  ready  method  of  drawing  a  diam 
eter  conjugate  to  a  given  diameter.  Thus  let  PC,  Fig.  223,  be  the  given 
diameter.  Circumscribe  the  circle,  produce  the  ordinate  PE  to  D,  draw 
DA,  and  D'A  perpendicular  to  it.  From  D'  let  fall  the  perpendicular 
D'E',  and  P'is  the  vertex  of  the  conjugate  diameter  required. 

341.  JProp. — The  rectangle  of  the  radii  vector es  drawn  to  the  ex 
tremity  of  any  diameter  equals  the  square  of  the  semi-conjugate  diameter. 

DEM. — Let  F'P  =  r',  and  PF  =  r,  Fig.  223,  and  the  other  notation  remain 
as  before.  Then  from  F'PE  we  have  r'  =  \/y'z  -|-  (Ae  -j-  x,*  = 


-  x-)(l  —  eu)  -f-  -42«2  -f-  tAex  -{-  x«  =  V A"  -\-  '2Aex  -J-  e-x*  —  A  -f-  ex.  In 
like  manner  from  P  E  F,  r  =  A  —  ex.  Whence  rr'  =  A-  —  e-x2.  Again  PA  = 
y , 2  -f-  Xj 2  =  B-  sin2  cp'  -f-  A*  cos2  <p'  =  (^.2  —  A-e2)  sin2  <p'  -f-  A-  cos  "<p'  =  ./I*  — 
^4.?e2sin2  cp'.  But  9>'  =  90°  -f-  cp ;  whence  smcp'  =  coscp,  and  P'A  =  A- — 
e2  •  ^I2cos2  cp  =  A~  —  e=x2.  .-.  rr'  =  P'A.  Q.  E.  D. 

342.  IP  Top. — The  sum  of  the  squares  of  any  pair  of  conjugate 
diameters  is  constant  and  equal  to  the  sum  of  the  squares  of  the  axes. 

DEM.— In  Fig.  223  we  have  P'A*  =  x,2  -f-  yj-  —  J^sin2^)'  -j-  A'  cos2  cp'  ;  or 
since  cp'  =90°-}-  <*,  sin  cp'  =  cos  cp,  and  cos  cp'=  —  sin  cp, 

P'A   =  A-  sin2  cp  -{-  B-  cos2  cp  ;  and  in  like  manner, 
PA2  =  A* cos2  cp  -f  J?*  sin2  <p. 

Adding  "PA*  +  P^A2  =  A*  -f  JS*.  Multiplying  by  4, 

4PA2  +  4P  A2  =  4 A2  -f  4^2.     Q.  E.  D. 

SCH. — This  proposition  has  been  demonstrated  before  (318,  a),  but  is 
inserted  here  as  its  demonstration  affords  an  example  of  the  utility  of  the 
eccentric  angle. 

Ex.  1.  "What  is  the  eccentric  angle  of  the  extremity  of  the  trans- 


SPECIAL  PROPERTIES  OF  THE  CONIC  SECTIONS. 


201 


yerse  axis  ?     What  of  the  extremity  of  the  latus  rectum  ?     What  of 
the  extremity  of  the  conjugate  axis  ? 

Ans.,  <p  =  0°,  cp  =  cosr\e  =  sin"1  — ,  <p  =  90°. 

Ex.  2.  In  an  ellipse  whose  axes  are  8  and  6,  what  is  the  eccentric 
angle  at  x  —  1  ?  What  are  the  co-ordinates  of  the  point  of  which 
the  eccentric  angle  is  60°  ?  45°  ?  30°  ? 

Ex.  3.  In  an  ellipse  whose  axes  are  12  and  8  what  is  the  length  of 
the  diameter  from  the  point  whose  eccentric  angle  is  60°  ? 


Suo. — Calling  the  semi-diameter  A2  we  have 
X  (i)2  -4-  16  X  (iv/3)*  =  21,  and  At  =  v/21. 


A2  cos2  <p  -|-  -B2  sin2  <p  = 


343.  Prop. — The  intercepts  of  a  secant  between  the  hyperbola  and 
its  asymptotes  are  equal. 

DEM. — Let  DD'  be  any  secant,  and  P 
and  P',  the  points  iri  which  it  cuts  the 
curve,  be  designated  respectively  as  (V,  y') 
and  (x",  y").  Since  DD'  is  a  line  pass 
ing  through  the  two  points  (x,  y'\  and 
(x",  y"},  we  have  for  its  equation  y  —  y'  = 

y ,       y-(x  —x).     And  since   (x ',  y')   and 

X    —  X 

(#",  y"}  are  points  in  the  curve  x'y  = 
x"y"  =  m.  If  in  the  equation  of  DD' 
we  make  y  =  0,  x  =  A  D,  and  x  —  x  = 

CD.      Hence  we  have   CD  =  x  —  x    =  ~  *„      "7   =':L2-. ^~  =  x" 

y    —y 

C'P'.     Now  as   PCD  and  P'C'D'  are  equiangular  and  have  CD  =  C'P' 
the  triangles  are  equal,  and  PD  =  P'.D'.     Q.  E.  ix 

SCH. — This  proposition  affords  an  ele 
gant  and  convenient  method  of  construct 
ing  the  hyperbola.  If  the  axes  are  given, 
put  them  in  position  and  draw  the 
asymptotes,  which  are  the  diagonals  of 
the  rectangle  on  the  axes.  Then,  through 
the  extremities  of  the  transverse  axis, 
draw  a  convenient  number  of  radiant 
lines,  as  aa',  bb',  cc',  dd',  and  make  the 
intercepts  la',  2Z>',  3c',  4oT  respectively 
equal  to  Ba,  Bb,  Be,  Bd.  Then  are  1,  2, 
3,  4,  points  in  the  curve. 

If  the  asymptotes   are    giyen,  or  the 


Fio.  225. 


202  PROPERTIES   OF   PLANE   LOCI. 

angle  included,  and  any  point  in  the  curve  as  P,  the  asymptotes  can  be 
drawn  ;  and  then  radiant  lines  through  P  will  be  secants  whose  intercepts 
will  make  known  points  in  the  curve. 

PARAMETER  TO  ANY  DIAMETER. 

344.  A  Parameter  to  any  Diameter  of  an  Ellipse  or 
Hyperbola,  is  a  third  proportional  to  that  diameter  and  its  conjugate. 
In  the  Parabola  it  is  a  third  proportional  to  any  abscissa  and  its  cor 
responding  ordinate. 

34 J.  Prop. — The  distance  from  any  point  in  a  Parabola  to  the 
focus  is  one  fourth  the  parameter  to  tJie  diameter  from  that  point. 

DEM.— Let  A2  F  =/ ;  there  is  y2*  =  4fxt.     From  Me.  12,  page  88,  we  have?/22 
—  --r^— ,xt ;  |md  also  2n  sin  a'  —  2p  cos  a'  =  0. 
From  the  latter,  n2  sin2o:'  =  p2  cos'^o:'  =  p2  — 
p2sin2a';   whence  sin2a'  =  —  j— — .      Hence 

2p  2(n2-f  p^  __  2(2pm  +  p2)  _ 

sin-  a'    "  p  p 

4(m  -|-  \p),  since  n2  =  2pm.  But  m  +  ip  = 
TF  =  FA2  +  /.  Therefore  t/22  =  4fxt, 
or  xt  :  t/2  : :  yt  :  4/ ;  and  4/  is  the  parameter 
to  the  diameter  A2jc,  Q.  E.  ». 

346.  COR.  1. — The  parameter  to  any  diameter  of  a  Parabola  is  four 
times  the  distance  from  the  vertex  of  that  diameter  to  the  directrix. 

347.  COB.  2. — The  double  ordinate  to  any  diameter  of  a  Parabola, 
which  (ordinate)  passes  through  the  focus,  is  the  parameter  to  that  dia 
meter. 

DEM.— Let  A2H  =  x2,  and  LH  =  yt;  whence  y22  =  4fxt.  Now  A2H  = 
xt  —  T-F  =  A2  F  =/.  Wherefore  t/22  =  4/2  ;  and  y2  =  2/.  But  I  L  = 
2LH  =  2y2  =  4/.  .-.  I  L  is  the  parameter  to  A2x,. 


348.  JProp. — Any  chord  which  passes  through  the  focus  of  an 
Ellipse  is  a  third  proportional  to  the  transverse  axis  and  a  diameter  par 
allel  to  the  chord. 

DEM.— Let  PF  =  r,  PFB  =  a,  and  P'F  =  r'.  Then 

r= 1- and  r'  = — -¥ (107);  whence  r 

1  —  e  cos  a  1  -f-  e  cosa  J 

Q«  C 

But  from    Ex.  10,   page 


A*  sm2a  -f-  B*  cos2a       "   1  —  e2  cos-o: 


SPECIAL  PROPERTIES  OF  THE  CONIC   SECTIONS.  203 

r £  t    a    ;  by  substituting  A*(l  —  e2)  for  B2  and  reducing.    Therefore—-  = 

-f-,  or  2  A:  B.C,  :  :  B1C1  :  PP ".     Q.  E.  D. 

A. 

349.  Sen. — The  statement  in  (347)  is  not  true  in  case  of  the  ellipse,  as 
will  appear  from  this  proposition. 

END    OF   PART   FIRST. 


GENERAL  SCHOLIUM. 

Book  Second,  treating  of  Loci  in  Space,  is  reserved  for  a  second  volume. 
The  present  volume  is  deemed  sufficient  for  the  use  of  all  students  in  our 
colleges,  except  such  as  pursue  mathematical  studies  as  a  specialty.  Vol 
ume  II.  will  contain  Loci  in  Space,  and  a  more  extended  course  in  the 
Calculus. 


THE 


INFINITESIMAL 
CALCULUS. 


THE 

INFINITESIMAL  CALCULUS. 


INTRODUCTION. 

[NOTE.— The  four  following  chapters  on  the  Differential  Calculus  are  to  be  read  immediately 
after  the  first  three  chapters  of  the  General  Geometry,  that  is,  the  first  92  pages  of  this  volume.] 

1.  Quantity  is  the  amount  or  extent  of  that  which  may  be 
measured  ;  it  comprehends  number  and  magnitude.  (See  ART.  4, 
General  Geometry,  and  the  two  Scholiums  under  it  on  pages  1 
and  2.) 

2»  Number  is  quantity  conceived  as  made  up  of  parts,  and 
answers  to  the  question,  "How  many?"  (See  ART.  5,  Illustration, 
General  Geometry.) 

3.  Number  is  of  two  kinds,  Discontinuous  and  Contin 
uous. 

4.  Discontinuous  Number  is  number  conceived  as  made 
up  of  finite  parts ;  or  it  is  number  which  passes  from  one  state  of 
aggregation  to  another  by  the  successive  additions  of  finite  units, 
i.  e,,  units  of  appreciable  magnitude. 

5.  Continuous  Number  is   number  which  is  conceived  as 
composed  of  infinitesimal  parts  ;   or  it  is  number  which  passes  from 
one  sta^e  of   value  to  another  by  passing   through  all  intermediate 
values,  or  states. 

ILL'S.— The  method  of  conceiving  number  with  which 
the  pupil  has  become  familiar  in  arithmetic  and  algebra, 
characterizes  discontinuous  number.  Thus  the  number 
13  is  conceived  as  produced  from  5  by  the  successive  ad 
ditions  of  finite  units,  either  integral  or  fractional.  In 
either  case  we  advance  by  successive  steps  of  finite  length- 
If  we  say  5,  6,  7,  etc.,  till  we  reach  13,  we  pass  by  one  FlG-  !• 

kind  of  steps  ;  and,  if  we  say  5.1,  5.2,  5.3,  etc.,  till  we  reach  13,  we  pass  by 
another  sort  of  steps  (tenths),  but  as  really  "by  finite  ones.  If,  however,  we  call  the 
line  AB,  Fig.  1,  x,  and  CD,  x',  and  conceive  AB  to  slide  to  the  position  CD, 
increasing  in  length  as  it  moves  so  as  to  keep  its  extremities  in  the  lines  O  M  and 


2  INFINITESIMAL   CALCULUS. 

O  N,  it  will  pass  by  infinitesimal  elements  of  growth  from  the  value  x,  to  the  value 
x'  ;  or,  it  will  pass  from  one  value  to  the  other  by  passing  through  all  intermediate 
values,  and  thus  becomes  an  illustration  of  continuous  number. 

Again,  if  the  line  AB,  Fig.  2,  be  considered  as  gen-       , . 

erated  by  a  point  moving  from  A  to  B,  and  we  call 

the  portion  generated  when  the  point  has  reached  C,  G' 

x,  and  the  whole  line  x',  x  will  pass  to  x  ,  by  receiving  infinitesimal  increments, 

or  by  passing  through  all  states  of  value  between  x  and  x'. 

A  surface  may  be  considered  as  generated  by  the  mo-     A          c  E 

tion  of  a  line,  and  thus  afford  another  illustration  of 
continuous  number.  Thus  let  the  parallelogram  AF 
be  conceived  as  generated  by  the  right  line  A  B  moving 
from  AB  to  EF.  When  AB  has  reached  the  po- 
sition  CD,  call  the  siirface  traced,  namely  A  BCD, 

jc,  and  the  entire  surface  A  B  E  F,  a;'  ;  then  will  x  pass  to  x  by  receiving  infinites 
imal  increments,  or  by  passing  through  all  intermediate  values. 

Finally,  as  volumes  may  be  conceived  as  generated  by  the  motion  of  planes,  all 
geometrical  magnitudes  afford  illustrations  of  continuous  number. 

We  usually  conceive  of  time  as  discontinuous  number,  as  when  we  think  of  it  as 
made  up  of  hours,  days,  weeks,  etc.  But  it  is  easy  to  see  that  such  is  not  the 
way  in  which  time  actually  grows.  A  period  of  one  day  does  not  grow  to  be  a 
period  of  one  week  by  taking  on  a  whole  day  at  a  time,  or  a  whole  hour,  or  even 
a  whole  second.  It  grows  by  imperceptible  increments  (additions).  These  incon 
ceivably  small  parts  of  which  continuous  number  is  made  up  are  called  Infinites 
imals. 

Motion  and  force  afford  other  illustrations  of  continuous  number.  In  fact,  the 
conception  which  regards  number  as  continuous,  will  be  seen  to  be  less  artificial — 
more  true  to  nature — than  the  conception  of  it  as  discontinuous. 

6.  An  Infinite  Quantity  is  a  quantity  conceived  under  such 
a  form,  or  law,  as  to  be  necessarily  greater  than  any  assignable  quan 
tity. 

7.  An  Infinitesimal  is  a  quantity  conceived   under  such  a 
form,  or  law,  as  to  be  necessarily  less  than  any  assignable  quantity. 

8.  Sen. — By  an  infinite  quantity  is  not  meant  one  larger  than  any  other, 
or  the  largest  possible  quantity.     It  simply  means  a  quantity  larger  than 
any  assignable  quantity  ;  i.  e. ,  larger  than  any  one  which  has  limits.     The 
mathematical  notion  concerns  rather  the  manner  of  conceiving  the  quantity, 
than  its  absolute  value.     Thus,  a  series  of  Is,  as  1  1  1,  etc.,  repeated  with 
out  stopping,  represents  an  infinite  quantity,  because,  from  the  method  of 
conceiving  the  quantity,  it  is  necessarily  greater  than  any  quantity  which 
we  can  assign  or  mention.     If  we  assign  a  row  of  9s  reaching  around  the 
world,  though  it  is  an  inconceivably  great  number,  it  is  not  as  great  as  a 
series  of  Is  extending  without  limit.     Moreover,  one  infinite  may  be  larger 
than  .another  ;  for  a  series  of  2s  extending  without  limit,  as  2  2  2  2,  etc.,  is 


INTRODUCTION.  3 

twice  as  large  as  a  series  of  Is  conceived  in  the  same  way.  It  is  never  of 
any  use  to  try  to  comprehend  the  magnitude  of  an  infinite  quantity  ;  we 
cannot  do  it ;  although  we  can  compare  infinites  just  as  well  as  finites. 

Again,  and  what  is  more  to  our  purpose,  an  infinitesimal  quantity  is  not 
a  quantity  so  small  that  there  can  be  no  smaller.  There  would  be  but  one 
such  quantity  and  hence  no  comparison  of  infinitesimals.  All  that  is  meant 
by  the  term,  as  used  in  mathematics  is,  a  quantity  which  is  to  be  treated 
in  the  argument  as  less  than  any  assignable  quantity.  Whether  we  can 
or  cannot  comprehend  its  absolute  magnitude  is  of  no  manner  of  con 
sequence.  Nor  is  absolute  value  usually  of  any  importance  in  pure  mathe 
matical  reasoning.  Thus  2  times  5  is  10  whether  5  be  mites  or  moun 
tains.  In  order  to  free  himself  from  needless  embarrassment  in  the  use 
of  infinitesimals,  the  student  needs  to  keep  constantly  in  mind  the  fact  that, 
In  pure  mathematics,  it  is  the  relation  of  quantities,  rather 
than  their  absolute  values,  with  which  we  are  concerned. 


0*  J?rop. — Any  finite  quantity  divided  by  an  infinite  is  an  infinites 
imal  ;  and  any  finite  quantity  divided  by  an  infinitesimal  is  an  infinite. 

DEM. — Let  a  represent  any  finite  quantity  and  x  any  infinite.  Then  -  is  an  in 
finitesimal  ;  for  the  value  of  a  fraction  depends  upon  the  relative  values  of  its 
numerator  and  denominator,  and  is  less  as  the  ratio  of  numerator  to  denominator 
is  less.  Now,  in  this  case,  a  is  infinitely  less  than  x,  by  the  definition  of  an  infinite. 

Hence  -  is  an  infinitesimal.     Again  -  is  infinite  if  x  is  infinitesimal,  since  a  is  in- 

x  x 

finitely  greater  than  x. 

10.  Con. — The  reciprocal  of  an  infinite  is  infinitesimal,  and  the  re 
ciprocal  of  an  infinitesimal  is  infinite. 

11.  The  products  of  infinites  by  infinites,  and  of  infinitesimals  by 
infinitesimals  are  denominated  Orders:   thus,  if  x  and  y  are  in 
finites,  #2,  yz,  and  ccy  are  infinites  of  the  Second  Order ;  if  x,  y, 
and  z  are  infinites,  x3,  z3,  xyz,  x*y,  xy2,  etc.,  are  infinites  of  the  Third 
Order.     The  corresponding  expressions  are  used  with  reference  to 
infinitesimals,  the  product  of  two  infinitesimals  being  called  an  infin 
itesimal  of  the  second  order,  of  three,  the  third,  etc. 

12.  Sen. — An  infinite  of  a  lower  order  sustains  a  relation  to  the  next 
higher  similar  to  that  which  a  finite  sustains  to  an  infinite.     Thus  if  x  and  y 
are  infinites,  x2,  xy,  and  y-  are  infinitely  greater  than  x  and  y.     On  the  other 
Hand  if  x  and  y  are  infinitesimals,  as*,  xy,  and  y2  are  infinitely  less,  and  sus 
tain  a  relation  to  x  and  y,  similar  to  that  which  infinitesimals  sustain  to 
finites. 


INFINITESIMAL   CALCULUS. 


AXIOMS. 

13.  From  expressions  containing  the  sum  or  difference  of  finites 
and  infinites,  the  finites  may  be  dropped  without  affecting  the  ratio. 

14.  From  expressions  containing  the  sum  or  difference  of  infin 
ites  of  different  orders,  the  terms  containing  the  lower  orders  may  be 
dropped  without  affecting  the  ratio. 

lo .  The  order  of  an  infinite  is  not  altered  by  multiplying  or  divid 
ing  it  by  a  finite. 

10.  From  expressions  containing  the  sum  or  difference  of  finites 
and  infinitesimals,  the  infinitesimal  terms  may  be  dropped  without 
affecting  the  ratio. 

17.  From  expressions  containing  the  sum  or  differenqe  of  infini 
tesimals  of  different  orders  the  terms  containing  the  higher  orders 
may  be  dropped  without  affecting  the  ratio. 

15.  The  order  of  an  infinitesimal  is  not  changed  by  multiplying 
or  dividing  it  by  a  finite. 

ILL'S.— Although  the  above  are  conceived  to  be  axioms  in  the  strictest  sense, 
that  is  truths  to  which  the  mind  at  once  assents  as  soon  as  the  terms  used  are 
clearly  comprehended,  the  true  notion  of  infinites  and  infinitesimals  is  so  removed 
from  common  thought  that  a  familiar  illustration  or  two  may  aid  the  comprehen 
sion.  Suppose,  then,  that  the  quantities  under  consideration  were  the  masses  of 
matter  in  the  earth  and  in  the  sun.  If  a  grain  of  sand  were  added  to  or  subtracted 
from  each  or  either  it  would  not  appreciably  affect  the  ratio  of  these  masses.  But 
in  this  instance  the  grain  of  sand  is  by  no  means  infinitesimal  with  reference  to 
either  mass  ;  it  is  &  finite,  though  very  small  part,  of  either  mass. 

Again,  let  x  and  y  be  two  infinite  quantities,  and  a  and  6  two  finite  ones.     There 

can  be  no  difference  between  — =—  and  '-  ;  since  to  assume  such  a  difference 

y  ±  b  y 

would  be  to  assign  some  values  to  a  and  6,  as  respects  x  and  y.     But  by  hypoth 
esis,  the  former  have  no  assignable  values  in  relation  to  the  latter. 

Once  more,  if  a  and  b  are  finite  quantities,  and  x  and  y  infinitesimal,  ~  =  j, 
since  x  and  y  have  no  assignable  values  as  compared  with  a  and  6.  So  also,  x  and 
y  still  being  infinitesimal,  —  ^  =  -,  as  x2  and  y*  are  infinitesimals,  (have  no 


assignable  values)  with  respect  to  x  and  y. 


INTRODUCTION.  5 

EVALUATION    OF   EXPRESSIONS   CONTAINING   FINITES   AND 
INFINITESIMALS,   AND  FINITES  AND  INFINITES. 

2^;  (i 

Ex.  1.  What  is  the  value  of  the  fraction  7- r  if  x  is  infinite  and 

3x  +  b 

a  and  b  finite  ? 

SOLUTION. — Since  a  and  6  have  no  assignable  values  in  relation  to  x  they  must 

2?*  2 

be  dropped,  and  we  have  -^-.     Now  dividing  both  terms  by  x,  we  have  -  as  the 
o*C  o 

value  of when  a;  is  infinite  and  a  and  &  finite. 

3x  -\-  b 

Ex.  2.  What  is  the  value  of  the  fraction  in  the  last  example  if  x  is 
infinitesimal  and  a  and  6  finite  ? 

SOLUTION. — As  x  is  infinitesimal  2.r  and  3x  are  also  infinitesimal,  and  hence  have 
no  value  in  relation  to  a  and  6,  and  must  be  dropped.     Hence  the  value  of  the 

a 

fraction  is  —  -. 
o 

^  2,772 3# 

Ex.  3.  What  is  the  value  of  — •  when  x  is  infinite?     When 

%JC'2  X 

x  is  infinitesimal  ? 

Ans.,  When  x  is  infinite,  6  ;  when  infinitesimal,  3. 

5x 

Ex.  4.  What  is  the  value  of  y  in  the  equation  y  =  -j when  x 

i+j 

is  infinite  ?     When  x  is  infinitesimal  ? 

Ans.,  When  x  is  infinite,  —  5  ;  when  infinitesimal,  -. 

Ex.  5.  What  is  the  value  of  y  in  the  expression  y  = when  x 

J.  *T"  QC 

is  infinite  ?     When  x  is  infinitesimal  ? 

A  ns.t  When  x  is  infinite,  0  ;  when  infinitesimal,  1. 

o.r3  I  bx2  I  ex  I  d 

Ex.  6.  What  is  the  value  of  — : when  x  is  infinite? 

mx3  -\-  nxz  +  px  -f  q 

When  x  is  infinitesimal  ? 

-4ns.,  WTien  'x  is  infinite,  -  ;  when  infinitesimal,  -. 
m  q 

Ex.  7.  What  is  the  value  of  y  in  the  expression  y  — ' 

3x*  —  nix 

when  x  is  infinite  ?     When  x  is  infinitesimal  ? 

Ans.,  When  x  is  infinite,  0  ;  when  infinitesimal,  5m. 


6  INFINITESIMAL  CALCULUS. 

a.x'-<  -f-  2.T2 1 

Ex.  8.  What  is  the  value  of  y  in  the  equation  y  = 

mx3  —  Bar  -f  2 

when  x  is  infinite  ?     When  x  is  infinitesimal  ? 

Ans.,  When  x  is  infinite,  y  =  GO  ;  when  infinitesimal,  y  =  —  -^ . 

3.r 

Ex.  9.  When  x  and  T/  are  infinitesimals  what  is  the  value  of   --  ? 

By 

Ans.,  We  cannot  tell ;  as  we  know  nothing  about  the  relation  be 
tween  x  and  y. 


Ex.  10.  What  is  the  value  of  —  wrhen  7/2  =  9a?  and  x  and  y  are 
infinite?  Ans.,  oc. 

Ex.  11.  Same  as  Ex.  10,  only  x  and  y  infinitesimal?  Am.,  0. 

Ex.  12.  Wliat  is  the  value  of  y  in  the  equation  ?/-'  =  — '•  —,  when 
x  is  infinitesimal?  Ans.,  0. 

CONSTANTS  AND  VARIABLES. 

10.  A  Constant  quantity  is  one  which  maintains  the  same 
value  throughout  the  same  discussion,  and  is  represented  in  the  no 
tation  by  one  of  the  leading  letters  of  the  alphabet. 

20.  Variable  quantities  are  such  as  may  assume  in  the  same 
discussion  any  value,  within  certain  limits  determined  by  the  nature 
of  the  problem,  and   are    represented  by  the  final    letters  of  the 
alphabet. 

21.  COR. — Any  expression  containing  a  variable  is,  when  taken  as  a 

whole,  a  variable.     Thus  the  value  of -the  ENTIRE  expression  (4a — 3#2  +  5) 5 
varies  if  x  varies ;  so  that  TAKEN  AS  A  WHOLE  it  is  a  variable. 

[NOTE. — These  notions  should  be  already  familiar  from  General  Geometry,  page  9,  and  are  in 
troduced  here  only  to  give  completeness,  and  for  review.] 

22.  Variables  are  distinguished  as  Independent  and  Dependent. 

23.  An  Independent  Variable  is  one  to  which  we  assign 
arbitrary  values,  or  upon  whose  law  of  variation  we  make  some  arbi 
trary  hypothesis. 

24.  A  Dependent  Variable  is  one  which  varies  in  value  in 
consequence  of  the  variation  of  the  independent  variable  or  vari 
ables. 

ILL. — Thus,  in  the  equation  of  the  parabola,  ?/2  =  2px,  if  we  assign  arbitrary 


INTRODUCTION. 


FIG.  4. 


values  to  x  and  find  the  corresponding  values  of  y,  we  make 
x  the  independent  variable,  and  y  the  dependent  variable. 
Again,  and  what  is  more  to  our  present  purpose,  if  we  as 
sume  x  to  vary  in  some  particular  way,  as  by  taking  on  equal 
increments,  as  DD',  D'D",  D"D'",  etc.,  y  will  evi 
dently  vary  in  some  other  way,  but  still  in  a  way  depending 
upon  the  way  in  which  x  varies,  and  upon  the  nature  of  the 
curve,  or,  what  is  the  same  thing,  upon  the  form  of  the 
equation  of  the  curve.  In  this  case  also,  x  is  the  independ 
ent  and  y  the  dependent  variable. 

gCH> This  distinction  is  made  simply  for  convenience,  and  is  not  founded 

in  any  difference  in  the  nature  of  the  variables ;    either  variable  may  be 
treated  as  the  independent  variable. 

2 £•  An  Equicrescent  variable  is  one  which  is  assumed  to  in 
crease  or  decrease  by  equal  increments  or  decrements,  as  x  in  the  last 
illustration. 

26.  Contemporaneous  Increments  are 

such  as  are  generated  at  the  same  time. 


ILL. — Thus  let  y  =/(»)  represent  the  equation  of  A  M  in 
the  figure.  Suppose  we  contemplate  the  values  of  x  and  y 
at  the  point  P'.  Now  if  x  takes  the  increment  D'D",  y 
takes  the  contemporaneous  increment  P"E'.  So  also  we 
see  that  DD',  P'E,  and  PP'  are  contemporaneous  incre 
ments  of  the  abscissa,  ordinate,  and  arc,  respectively. 


PF 

P/HE 


D  D'  D"     X 


FIG.  5. 


FUNCTIONS   AND   THEIR   FOKMS. 

2V.  A  Function  is  a  quantity,  or  a  mathematical  expression, 
conceived  as  depending  for  its  value  upon  some  other  quantity  or 
quantities. 

ILL. A  man's  wages  for  a  given  time  is  a  function  of  the  amount  received  per 

day ;  or,  in  general,  his  wages  is  a  function  of  both  the  time  of  service  and  the 
amount  received  per  day.  Again,  in  the  expressions  y  =  2ax9,  y  =  x3  —  2&x  -f  5, 
y  =  2  log  ax,  y  ==  a*,  y  is  a  function  of  x  ;  since,  the  numbers  2,  5,  a  and  6  being 
considered  constant,  the  value  of  y  depends  upon  the  value  we  assign  to  x.  For 
a  like  reason  v/a2  —  &,  and  3a»*  —  2v/ie  may  be  spoken  of  as  functions  of  x. 
Once  more,  the  ordinate  of  a  curve  is  a  function  of  the  abscissa. 

gCH There  is  a  sense  in  which  the  dependent  variable  (or  function)  is  a 

function  of  the  constants  as  well  as  of  the  variable  or  variables  which  enter 
into  its  value.  So  also  it  is  a  function  of  the  form  of  the  expression,  that 
is,  its  value  depends  in  part  upon  the  form  of  the  expression  as  well'  as 
upon  the  value  of  the  independent  variable.  Thus  if  we  have  y  =  a 


8  INFINITESIMAL   CALCULUS. 

+  b,  and  y  =  #3  —  ex,  though  in  each  case  y  is  a  function  of  x,  speaking 
according  to  the  definition,  nevertheless  it  is  not  the  same  function  in  both 
cases.  Its  value  depends  upon  the  value  of  x,  upon  the  constants,  and 
upon  the  form  of  the  expression  involving  these  quantities.  But  the  con 
ception  expressed  in  the  definition  is  the  ordinary  one. 

28.  Functions  are  classified  by  their  forms  as  Algebraic  and 
Transcendental,  and  the  latter  are  subdivided  into  Trigono 
metrical  and  Circular,  Logarithmic  and  Exponential. 

29.  An  Algebraic  Function  is  one  which  involves  only  the 
elementary  methods  of  combination,  viz.,  addition,  subtraction,  mul 
tiplication,  division,  involution  and  evolution.     Thus  in  y=  ax- — 3#3, 
y  is  an  algebraic  function  of  x. 

30.  A  Trigonometrical  Function  is  one  which  involves 
sines,  cosines,  tangents,  cotangents,  etc.,  as  variables  ;  thus  ?/— sintf, 
y  =  sin  x  tan  xy  etc. 

31.  A   Circular   Function  is  one  in  which  the  concept  is  a 
variable  arc  (in  the  trigonometrical  the  concept  is  a  right  line).    These 
are  written  thus  :  y  =  sin~X  read  "y  equals  the  arc  whose  sine  is  x "; 
y  =  tan"1^,  read  " y  equals  the  arc  whose  tangent  is  x" 

ILL. — Notice  that  in  the  expression  y  =  tan— lx,  it  is  the  arc  which  we  are  to 
think  of,  while  in  the  expression  x  =  tan?/  it  is  the  tangent,  which  is  a  right  line. 
Trigonometrical  functions  are  right  lines  ;  circular  functions  are  arcs.  These 
functions  are  mutually  convertible  into  each  other  ;  thus  y  =  sin—'  x  is  equivalent 
to  £  =  sin  y,  the  only  difference  being  that  in  the  former  we  think  of  the  arc,  the 
sine  being  given  to  tell  what  arc,  and  in  the  latter,  we  think  of  its  sine,  the  arc 
being  given  to  tell  what  sine. 

The  circular  functions  y  =  sin~~lj?,  y  =  cos~X  y  =  sec~lx,  etc.,  are 
often  called  Inverse  Trigonometrical  Functions. 

32.  A  Logarithmic  Function  is  one  which  involves  loga 
rithms  of  the  variable  ;  as  y  =  loga?,  log2?/  =  Slogan,  etc. 

33.  An  Exponential  Function  is  one  in  which  the  vari 
able  occurs  as  an  exponent ;  as  y  =  a*,  z  =  xv>  etc. 


34.  Functions  are  further  distinguished  as  Explicit  and  Im 
plicit. 

35.  An  Explicit*  Function  is  a  variable  whose  value  is  ex 
pressed  in  terms  of  another  variable  or  other  variables  and  constants. 

Thus  in  y  =  2ax3  —  So:"3",  y  is  an  explicit  function  of  x. 

*  From  explicitum,  unfolded.     The  function  is  disentangled  from  the  other  quantities. 


INTRODUCTION.  9 

36.  An  Implicit*  Function  is  a  variable  involved  in  an 
equation  which  is  not  solved.     Thus  in  x*  —  3xy  -f  2y  =  16,  y  is  an 
implicit  function  of  x,  or  x  is  an  implicit  function  of  y.     When  we 
can  solve  the  equation,  an  implicit  function  may  always  be  expressed 
as  explicit. 

37.  Notation.     "When  we  wish  to  write  that  y  is  an  explicit 
function  of  x,  and  do  not  care  to  say  precisely  what  the  form  of  the 
function  is,  we  write  y  =f(x),  read  "y  =  a  function  of  x."     If  we 
wish  to  indicate  several  different  forms  of  dependence  in  the  same 
discussion,  we  use  other  letters,  as  y=f(x),  y=F(x\  y=  (p(x),  etc., 
or  use   subscripts  or   accents  as  y  =fl(x),  y=f>(x),  etc.      Such 
symbols  are  read  "t/  =  the/,  large  F,   q>,  f  sub-one,  /  prime,  etc., 
function  of  x,"  as  the  case  may  be. 

When  we  wish  to  write  that  x  and  y  are  functions  of  ea.ch  other,  or 
that  y  is  an  implicit  function  of  x,  or  x  an  implicit  function  of  y, 
without  being  more  specific,  we  write  F(x,  y}  =  0,  or  f(x,  y)  =  0, 
or  (p(x,  y)  =  0,  etc.  ;  and  read  "function  x  and  y  =  0,"  the  F  func 
tion  x  and  y  —  0,  etc.  This  form  symbolizes  any  equation  between 
two  variables  with  all  the  terms  transposed  to  the  first  member. 


38.  Again,   functions   are   distinguished   as  Increasing   and 
Decreasing. 

39.  An  Increasing  Function  is  a  function  that  increases 
as  its  variable  increases,  and  decreases  as  its  variable  decreases. 

40.  A  Decreasing  Function  is  a  function  which  decreases 
as  its  variable  increases,  and  increases  as  its  variable  decreases. 

ILL. — In  the  expressions  y2  =  %px,  y  =  logic,  y  =  ax,  y  is  an  increasing  function 
of  x.  In  the  expressions  y  =  — ,  7/2  -f-  #2  =  JR2,  y  =  log  — ,  y  is  a  decreasing  func 
tion  of  x.  For  what  values  of  x  is  y  an  increasing  function  of  its  variable,  and 
for  what  a  decreasing,  in  the  following  :  y3  =  ay?  —  x2,  y  =  sinx,  y  =  cosx? 


41.  TJie  Infinitesimal  Calculus  treats  of  Continuous 
Number,  and  is  chiefly  occupied  in  deducing  the  relations  of  the  con 
temporaneous  infinitesimal  elements  of  such  number  from  given  re 
lations  between  finite  values,  and  the  converse  process,  and  also  in 
pointing  out  the  nature  of  such  infinitesimals  and  the  methods  of 
using  them  in  mathematical  investigation. 

*  From  implicitum,  infolded,  entangled.          * 


10 


INFINITESIMAL   CALCULUS. 


ILL. — Let  ?/2  =  8x  be  the  equation  of  the  parabola  in  the 
figure.  Here  we  have  the  relation  between  finite  values  of 
y  and  x  expressed.  Now  suppose  x  takes  an  infinitesimal 
increment  as  D  D '  *,  what  increment  does  y  take  ?  The  cal 
culus  shows  us  that  the  increment  which  y  takes  is  -  times 

y 

as  large  as  the  increment  which  x  takes  ;  that  is,  it  shows  us 
the  relation  between  the  elements  of  the  variables  y  and  x, 
when  we  know  the  relation  between  finite  values.  This  is 
'the  province  of  the  Differential  Calculus.  The  converse  of 


FIG.  6. 


this  problem  is,  What  is  the  equation  of  the  curve  whose  ordinate  varies  -  times 

y 

as  fast  as  its  abscissa  ?  that  is,  having  given  the  relation  between  the  infinitesimal 
elements  of  y  and  x,  to  find  the  relation  between  finite  values.  This  is  the  prov 
ince  of  The  Integral  Calculus. 

42.  There  are  two  branches  of  the  Calculus,  viz.,  The  Differ 
ential  Calculus,  and  The  Integral  Calculus. 


*  Of  course  all  such  attempts  to  represent  infinitesimals  to  the  eye,  are  egregious  exaggerations- 
nevertheless  they  axe  of  great  service  to  the  mind. 


THE 


INFINITESIMAL   CALCULUS 


CHAPTEE  I, 

THE  DIFFERENTIAL    CALCULUS. 


SECTION  I. 

Differentiation  of  Algebraic  Functions, 

43.  The  Differential  Calculus  is  that  branch  of  the  Infin 
itesimal   Calculus   which    treats  of    the    methods   of    deducing   the 
relations  between  the  contemporaneous  infinitesimal  elements  of  vari 
ables,  from  given  relations  between  finite  values. 

44.  A  Differential  is  the  difference  between  two  consecutive 
states  of  a  function,  or  variable.     It  is  the  same  as  an  infinitesimal. 

43.  Consecutive  Values  of  a  function  or  variable  are  values 
which  differ  from  each  other  by  less  than  any  assignable  quantity. 

Consecutive  Points  on  a  line  are  points  nearer  to  each  other  than 
any  assignable  distance. 

ILL. —Suppose  y  =  2x2  —  3x.  Now  let  x  be  supposed  to  increase  infmitesimally, 
y  will  also  change  infinitesimally.  Call  the  new  value  of  y,  y  .  Then  y  =  2x's 
3z'.  In  such  a  case  x  and  x'  are  consecutive  values  of  the  variable,  and  y  and  y' 
are  consecutive  values  of  the  function.  But  by  this  we  do  not  mean  that  x  and  x' 
(or  y  and  y' )  are  so  nearly  equal  that  there  can  be  no  intermediate  value,  for  this  would 
be  to  make  an  infinitesimal  mean  a  quantity  so  small  that  there  can  be  no  smaller, 
which  is  not  its  meaning  as  used  in  mathematics  (7).  All  that  is  meant  by  saying 
that  y  and  y'  are  consecutive  values  is  that  they  are  to  be  reasoned  upon  as  having 
no  assignable  difference. 

So  also  in  speaking  of  consecutive  points  on  a  line,  as  D  and  D',  or  P  and  P', 
FUj.  6,  we  do  not  conceive  them  as  actually  in  juxtaposition  ;  but  we  mean  simply 
that  we  are  to  reason  upon  them  as  nearer  each  other  than  any  assignable  distance. 

46.  Notation.  The  differential  of  a  variable  (one  of  its  infini 
tesimal  elements)  is  represented  by  writing  the  letter  d  before  it. 


12  THE  DIFFERENTIAL  CALCULUS. 

Thus,  dx,  read,  "  differential  x"   Of  course  the  letter  d  is  not  to  be  con 
founded  with  a  factor  ;  it  is  simply  an  abbreviation  for  differential 

[CAUTION.— The  student  should  be  careful  and  not  allow  himself  to  read  such 
expressions  as  dy,  dx,  etc.,  by  merely  naming  the  letters  as  he  would  ay,  ax,  etc. 
The  former  should  always  be  read  "differential  y,"  " differential  x,"  etc.] 


RULES   FOR   DIFFERENTIATING   ALGEBRAIC    FUNCTIONS. 

47 •    RULE  1. To     DIFFERENTIATE    A    SINGLE   VARIABLE    SIMPLY    WRITE 

THE  LETTER  d  BEFORE  IT. 

DEM.— Let  us  take  the  function  y  —  x.  The  consecutive  state  of  the  variable 
is  x  -f-  dx.  Now  representing  the  change  in  y  which  is  produced  by  this  change 
in  x  by  dy  (dx  and  dy  being  the  contemporaneous  increments  of  the  variable  and 
the  function),  we  have 

1st  state  of  the  function, y  =  x, 

2nd,  or  consecutive  state, y  -f-  dy  =  x  -f-  dx. 

Subtracting  the  1st  from  the  2nd,  ~~         dy  =  dx,  which 

being  the  difference  between  two  consecutive  states  of  the  function  is  its  differen 
tial  (44).      Q.  E.  D. 

SCH. — This  rule  is  evidently  only  the  same  thing  as  the  notation  requires, 
and  its  formal  demonstration  would  be  unnecessary  except  for  the  purpose 
of  uniformity  in  treating  the  several  cases  of  differentiation. 

ILL.— Let  M  N  be  a  line  passing  through  the  origin  and  making  an  angle  of 
45^.  with  the  axis  of  x.  Its  equation  is  y  =  x.  Let  P  be  any  point  in  the  line, 
A  D  =  x,  and  PD  —  y.  Let  x  take  the  infinitesimal  increment  D  O'(dx),  then 
y  becomes  P'  D'.  Now  the  first  state  of  the  function  is 

P  D  =  A  D,  or  y  =  x, 

The  second  or  consecutive  state  is  PD  +  P'E  =  A  D  +  D  D',  or  y  +  dy  =  x+dx. 
Subtracting  we  have  P'  E  =  D  D',  or  dy  =  dx. 

Now  that  the  increment  of  y  (or  dy)  is  equal  to  the  in 
crement  of  x  (or  dx)  in  this  case  is  readily  seen  from 
the  figure  ;  for,  as  P'PE  =  45°,  P'E.  =  PE,  or  DD'. 
dy  =  dx,  then,  means  that  the  contemporaneous  incre 
ments  of  x  and  y  are  equal,  or  that  x  and  y  increase  at 

the  same  rate. 

*N 

FIG.  7. 

48.  R  ULE  2. — CONSTANT  FACTORS  OR  DIVISORS  APPEAR  IN  THE  DIFFER 
ENTIAL  THE  SAME  AS  IN  THE  FUNCTION. 

DEM.— Let  us  take  the  function  y  =  ax,  in  which  a  is  any  constant,  integral  or 
fractional.  Let  x  take  an  infinitesimal  increment  and  become  x  +  dx  ;  and  let  dy 
be  the  contemporaneous  increment  of  y,  so  that  when  x  becomes  x  -f  dx,  y  be 
comes  y  -f-  dy.  We  then  have 


DIFFERENTIATION   OF  ALGEBRAIC   FUNCTIONS.  13 

1st  state  of  the  function, y  =  ax  ; 

2nd,  or  consecutive  state, y  -f-  dy  =  a(x  +  dx)  =  ax  -f  adx. 

Subtracting  the  1st  from  the  2nd,  dy  =  adx, 

which  being  the  difference  between  two  consecutive  states  of  the  function  is  its 
differential  (44) .  Now  the  factor  a  appears  in  this  differential  just  as  it  was  in 
the  function.  Q.  E.  D. 

ILL.— Let  y  =  ax  be  the  equation  of  the  line  MN. 
P  D  and  P'  D'  representing  consecutive  ordinates,  D  D' 
represents  dx,  and  P'  E  represents  dy.  Here  it  is  evident 
that  P'E  =  a  X  DD';  for  from  the  triangle  PP'E  we 
have  P'E  ==  tanP'PE  X  PE.  ButtanP'PE  = 
tan  M  AX  =  a.  The  meaning  in  this  case  is,  therefore, 
that  the  ordinate  increases  a  times  as  fast  as  the  abscissa. 

If  a  =  1,  or  tan  45°,  the  ordinate  and  abscissa  increase  at  equal  rates  ;  if  a  <  1, 
t  e.,  if  the  angle  is  less  than  45°  the  ordinate  increases  more  slowly  than  the 
abscissa  ;  if  a  >- 1,  the  ordinate  increases  more  rapidly  than  the  abscissa. 

40.  RULE  3. — CONSTANT  TERMS  DISAPPEAR  IN  DIFFERENTIATING:    OR 

THE  DIFFERENTIAL  OF  A  CONSTANT  IS  0. 

DEM. — Let  us  take  the  function  y  =  ax  d=  &,  in  which  a  and  &  are  constants. 
Let  x  take  an  infinitesimal  increment  and  become  x  -f  dx  ;  and  let  dy  be  the  con 
temporaneous  increment  of  y,  so  that  when  x  becomes  x  -f-  dx,  y  becomes  y  -j-  dy. 
We  then  have 

1st  state  of  the  function, y  =  ax  ±  &  ; 

2nd,  or  consecutive  state, y  -f  dy  =  a(x  +  dx}  =h  6, 

or y  -\-  dy  =  ax  -f-  adx  ±  &. 

Subtracting  the  1st  state  from  the  2nd,  ~  dy  =  adx,  which 

being  the  difference  between  two  consecutive  states  of  the  function  is  its  differen 
tial  (44).  Now  from  this  differential  the  constant  term  ±  &  has  disappeared. 
We  may  also  say  that  as  a  constant  retains  the  same  value  there  is  no  difference 
between  its  consecutive  states  (properly  it  has  no  consecutive  states).  Hence  the 
differential  of  a  constant  may  be  spoken  of  (though  with  some  latitude)  as  0. 
Q.  E.  D. 

ILL.— Let  y  =  ax-\-b  be  the  equation  of  the  line  M  N. 
Now  the  relative  rates  of  increase  of  the  abscissa  and  or 
dinate,  that  is  the  relation  of  dy  to  dx,  is  evidently  not 
affected  by  &  which  is  A  B  ;  for,  if  we  were  to  draw  a  line  ^ 

through  the  origin  parallel  to  M  N ,  the  contemporane-   -ft 
ous  increments  of  its  co-ordinates  would  be  the  same  as 
those  of  M  N.     Again,  we  can  see  that  the  constant  term  pIG  9 

does  not  affect  the  differential,  i.  e.,  the  difference  between 

the  consecutive  states  of  y,  by  observing  that  these  two  states  are  represented  by 
PD  and  P'  D',  each  of  which  contains  the  constant  as  a  part  of  it,  whence  the 
difference  between  them  is  not  affected  by  it. 

50.  COR. — An  infinite  variety  of  functions  differing  from  each  other 
only  in  their  constant  terms  still  have  the  same  differential. 


DD'    X 


14  THE   DIFFERENTIAL   CALCULUS. 

51.  RULE  4. To   DIFFERENTIATE   THE   ALGEBRAIC  SUM  OF  SEVERAL  VA 
RIABLES,  DIFFERENTIATE   EACH  TERM  SEPARATELY  AND  CONNECT  THE  DIFFEREN 
TIALS  WITH  THE  SAME  SIGNS  AS  THE  TERM9. 

DEM.  — Let  u  =  x  -\-  y  —  z,  u  representing  the  algebraic  sum  of  the  variables 
x,  y,  and  —  z.  Then  is  the  differential  of  this  sum  or  du  =  dx  -f-  dy  —  dz.  For 
let  d.c,  dy,  and  dz  be  infinitesimal  increments  of  x,  y,  and  z  ;  and  let  du  be  the  in 
crement  which  u  takes  in  consequence  of  the  infinitesimal  changes  in  a;,  y,  and  z. 
We  then  have 

1st  state  of  the  function, u  =  x-\-  y  —  z  ; 

2nd,  or  consecutive  state, u  -f-  du  =  x  -f-  dx  -f-  y  +  dy  —  (z  -f  dz), 

or u  -)-  du  =  x  -f-  dx  -f-  y  -f-  dy  —  z  —  dz. 

Subtracting  the  1st  state  from  the  2nd,  du  =  dx  -}-  dy  —  dz.  Q.  E.  r>. 

ILL. — We  may  illustrate  this  by  conceiving  x  and  y  to  be  forces  acting  to  raise 
a  weight,  and  z  a  force  acting  to  prevent  the  raising,  u  being  the  aggregate  effect  of 
all,  i.  e.  their  algebraic  sum  (Complete  Algebra,  65).  Now  if  x,  y,  and  z  each  re 
ceive  an  infinitesimal  increment,  which  we  will  call  respectively  dx,  dy,  and  dz,  it 
is  evident  that  the  increment  of  lifting  force  is  dx  -J-  dy,  and  as  the  increment  of 
the  depressing  force  is  dz,  the  combined  effect  of  the  change  is  dx  -\-  dy  —  dz, 
which  is  the  change  in  u.  Moreover,  since  this  quantity  dx  -\-  dy  —  dz  is  the  ag 
gregate  of  a  finite  number  of  infinitesimals,  it  must  be  itself  infinitesimal.  Hence 
the  change  in  u  is  infinitesimal,  or  du. 

SCH. — It  is  important  to  notice  that  the  above  reasoning  is  entirely  inde 
pendent  of  the  relative  values  of  the  infinitesimals  dx,  dy,  and  dz.  These 
may  be  conceived  as  equal,  or  as  sustaining  any  finite  ratio  whatever  to 
each  other,  only  so  that  they  remain  infinitesimal. 

52.  RULE  5. — THE  DIFFERENTIAL  OF  THE  PRODUCT  OF  TWO  VARIABLES 

15  THE  DIFFERENTIAL  OF  THE  FIRST  INTO  THE  SECOND,  PLUS  THE  DIFFERENTIAL 
OF  THE  SECOND  INTO  THE  FIRST. 

DEM.—  Let  u  =  xy  be  the  first  state.  The  consecutive  state  is  u  +  du  = 
(x  -j-  dx)(y  -f-  dy)  =  xy  -f-  ydx  -j-  xdy  -f-  dxdy.  Subtracting  the  1st  state  from  the 
2nd,  or  consecutive  state,  we  have  du  =  ydx  -\-  xdy  -f-  dx  •  dy.  Now  ydx  and  xdy 
are  infinitesimals  of  the  1st  order,  and  dx  •  dy,  being  the  product  of  two  infinitesi 
mals,  is  of  the  2nd  order  and  must  be  dropped  (17).  Therefore  du  =  ydx  -f-  xdy. 
Q.  E.  D. 

ILL. — Let  u  represent  the  area  of  the  rectangle  A  BO  D,  x  = 
*  A  B,  and  y  =  *  BO.  Then  u  =  xy.  Let  Bb  represent  dx,  and 
Cv",dy.  Whence  BbCc'  =  *ydx,  Dt?Cc"  =  *xdy,  Cc'cc"  =* 
dx  •  dy,  and  du  =  *BbCc'  -f-  DcZCc"  -f-  Cc'cc".  Now  since 
cc'  is  infinitesimal  and  c'b  is  finite,  Cc'cc"  is  infinitesimal  with 
reference  to  B6Cc',  as  for  a  like  reason  it  is  with  reference  to 
D  /Cc"  ;  hence  it  is  to  be  omitted  as  having  no  assignable  value  with  reference  to 
them. 

Another  view  which  may  be  taken  of  this  is  to  consider  that  it  is  the  rate  at  which 

*  In  such  cases  =  signifies  "  is  represented  by,"  and  is  used  for  brevity. 


DIFFERENTIATION   OF  ALGEBRAIC   FUNCTIONS.  15 

the  rectangle  is  increasing  when  x  =  AB  and  y  =  BC,  not  the  amount  of  change 
in  the  area  after  x  and  y  shall  have  increased  more  or  less  :  in  other  words,  we  seek 
for  the  difference  between  consecutive  values  of  the  area.  Now  it  is  easy  to  see 
that  the  rate  at  which  the  rectangle  A  BCD  starts  to  increase,  depends  upon  the 
length  of  the  side  BC  (y)  and  the  rate  at  which  it  starts  to  move  to  the  right,  -J-  the 
length  of  DC  (.?)  and  the  rate  at  which  it  starts  to  move  upward.  Letting  dx 
represent  the  rate  at  which  A  B  starts  to  increase  (by  being  the  amount  which  it 
would  increase  in  an  infinitesimal  of  time),  and  dy  represent  in  like  manner  the 
rate  at  which  y  starts  to  increase,  we  readily  see  that  du  =  ydx  -(-  xdy  is  the  rate 
at  which  the  area  starts  to  increase.  Moreover,  we  see  that  this  is  equally  true 
whether  dy  —  dx,  or  whether  one  is  any  finite  multiple  of  the  other  ;  all  that  is 
necessary  being  that  both  be  infinitesimals  of  the  same  order. 

S3.  B  ULE  6.  —  THE  DIFFERENTIAL  or  THE  PRODUCT  OF  SEVERAL  VARIA 

BLES  IS  THE  SUM  OF  THE  PRODUCTS  OF  THE  DIFFERENTIAL  OF  EACH  INTO  THE 
PRODUCT  OF  ALL  THE  OTHERS. 


DEM.—  Let  u  =  xyz  ;  then  du  =  yzdx  -j- 
For  the  1st  state  of  function  is  ..........................  u  =  xyz, 

2nd,  or  consecutive  state,  ...........  .'  ----    u  +  du  =  (ss+dx)(y+dy)(z+ds), 

or  ..........   u+du—xyz+  yzdx  -f  xzdy  -f  xydz  +  xdydz  +  ydxdz  -f  zdxdy  +  dxdydz. 

Subtracting  and  dropping  infinitesimals  of  higher  orders  than  the  first  we  have 
du  =  yzdx  +  xzdy  +  xydz. 

In  a  similar  manner  the  rule  can  be  demonstrated  for  any  number  of  variables. 
Q.  E.  D. 

54.  RULE  7.  —  THE  DIFFERENTIAL  OF  A  FRACTION  HAVING  A  VARIABLE 

NUMERATOR  AND  DENOMINATOR  IS  THE  DIFFERENTIAL  OF  THE  NUMERATOR 
MULTIPLIED  BY  THE  DENOMINATOR,  MINUS  THE  DIFFERENTIAL  OF  THE  DENOMI 
NATOR  MULTIPLIED  BY  THE  NUMERATOR,  DIVIDED  BY  THE  SQUARE  OF  THE 
DENOMINATOR. 

DEM.—  Let  u  =  -  ;  then  is  du  =  y  X  ~  *—  .     For  clearing  of  fractions  yu  =  x. 
Differentiating  this  by  Kule  5,  udy  -f-  ydu  =  dx.     Substituting  for  u  its  value,  we 

TV/?/  vr^  —  yfiy 

have  '—  +  ydu  ==  dx.  Finding  the  value  of  du,  we  have  du  =  '•  -  -  —  -. 
Q.  E.  D. 

55.  COR.  —  The  differential  of  a  fraction  having  a  constant  numerator 
and  a  variable  denominator  is  the  product  of  the  numerator  with  its  sign 
changed  into  the  differential  of  the  denominator,  divided  by  the  square  of 
the  denominator. 

DEM.  _  Let  u  =  -.     Differentiating  this  by  the  rule  and  calling  the  differential 

0  —  adii       —  ady 
of  the  constant  (a),  0,  we  have  du  =  -  7—  -  =  -  —  .     Q.  E.  D. 


16  THE   DIFFERENTIAL   CALCULUS. 

Sen. — If  the  numerator  is  variable  and  the  denominator 'constant  it  falls 
tinder  Rule  2. 

5G.  RLLE  8. — THE  DIFFERENTIAL  OF  A  VARIABLE   AFFECTED  WITH  AN 

EXPONENT  IS  THE  CONTINUED  PRODUCT  OF  THE  EXPONENT,  THE  VARIABLE  WITH 
ITS  EXPONENT  DIMINISHED  BY  1,  AND  THE  DIFFERENTIAL  OF  THE  VARIABLE. 

DEM. — 1st.  When  the  exponent  is  a  positive  integer.— "Let  y  =  xm,  m  being  a  pos 
itive  integer  ;  then  dy  =  mxm-ldx.  For  y  =  x™  =  x  -  x  •  x  •  x  to  m  factors.  Now 
differentiating  this  by  Rule  6,  we  have 

dy  =  (xxx  torn  —  1  factors)  dx  -f-  (xxx  torn  —  1  factors)  dx  -f-  etc.,  to  m  terms, 
or  dy  =  xm~ldx  -f-  xm~ldx  -f-  xm~ldx  -f  etc.,  to  m  terms. 
.  • .  dy  =  mxm—ldx. 

2nd.    When  the  exponent  is  a  positive  fraction.— "Let  y  =  o>,  ™  being  a  positive 

fraction  ;  then  dy  =  —x~~  dx.  For  involving  both  members  to  the  nth  power 
we  have  yn  =  xm.  Differentiating  as  just  shown,  ny"—ldy  =  mxm~ldx.  Now 
from  y  =  x  »,  we  have  yn~l  =  x~iT~.  Substituting  this  in  the  last  form,  we  have 

nx~~dy  =  mxm-1dx,  or  dy  =  ™xm-1-  ~v~tZic  =  —  x~~ldx. 

n  n 

3rd.  When  the  exponent  is  negative.— -Let  y  =  x~n,  n  being  integral  or  fractional  ; 
then  dy  =  —  nx-^dx.  For  y  =  x~»  =  -.,  which  differentiated  by  Rule  7,  Cor., 

gives  dy~  —  —  '  =  —  nx~n-ldx.  All  three  of  which  forms  agree  with  the 
enunciation  of  the  rule.  Q.  E.  D. 

£7.  COR. — The  differential  of  the  square  root  of  a  variable  is  the  dif 
ferential  of  the  variable  divided  by  twice  the  square  root  of  the  variable. 

DEM.— Let  y  =  \/x  =  x' .  Differentiating  by  the  rule  we  have  dy  =  ix^dx  = 
i  ~i  3  dx 

|g      dX  =  -.       Q.  E.   D. 

2v/x 

SCH.— Special  rules  can  be  readily  made  for  other  roots,  but  it  is  un 
necessary.  The  square  root  is  of  such  frequent  occurrence  as  to  make  the 
special  process  expedient.  Of  course  the  general  rule  can  always  be  used, 
if  desired. 


EXERCISES. 

[Nora.— The  following  examples  are  designed  to  give  practical  skill  in  applying  the  rules  for 
differentiating  algebraic  functions.  The  student  should  not  advance  beyond  these,  till  he  has 
the  rules  firmly  fixed  in  memory,  and  can  apply  them  with  facility  to  all  forms  of  algebraic  func 
tions.] 

Ex.  1.  Differentiate  y  =  Qx  —  4.  dy  =  Qdx. 

QUERY.— What  three  rules  apply?  Be  careful  to  repeat  the  rules  in  applying 
them  to  the  solution  of  the  examples,  and  thus  render  them  familiar. 


DIFFERENTIATION   OF  ALGEBRAIC   FUNCTIONS.  17 

Ex.  2.  Differentiate  y  =  OG  +  3a<^*  +  3a^4  -f  a*. 

SOLUTION.—  The  differential  of  y  is  dy.  [Repeat  Rule  1.]  To  differentiate  the 
second  member  we  notice  1st,  that  it  consists  of  several  terms,  and  hence  proceed 
to  differentiate  each  term  separately.  [Repeat  Rule  4.  ]  a6  being  a  constant  term, 
disappears.  [Repeat  Ride  3.  ]  To  differentiate  3a4x2,  we  notice  1st  that  the  con 
stant  factor  3a4  will  be  a  factor  in  the  differential.  [Repeat  Rule  2.]  The  differ 
ential  of  <c2  is  2xd.x.  [Repeat  Rule  8.]  Hence  the  differential  of  3a4x2  is  Qa4xdx. 
[In  like  manner  proceed  with  the  other  terms,  giving  the  reason  for  each  step  by 
repeating  the  appropriate  rule.} 

Ex.  3.  Differentiate  u  —  2ax  —  3#2  -f-  abx*  —  5. 

Result,  du  =  (la  —  6x  +  3abx*)dx. 

Ex.  4.  Differentiate  y  =  3#4  —  2x  —  5m. 
Ex.  5.  Differentiate  u  =  ab  —  G#3  +  2ax. 
Ex.  6.  Differentiate  u  =  ax-y3. 

QUERIES.  —  What  is  the  most  general  feature  of  the  function  ax^y3?  What  rulo 
applies  first?  Rule  5.  What  other  rule  applies  ? 

Result,  du  =  2axifdx  -f  Sax^fdy. 
Ex.  7.  Differentiate  u  =  6ax3y3. 

Result,  du  = 


Ex.  8.  Differentiate  y  =  2bz~*  +  &wr. 

21  SaAz       £bdz 

liesulL  dii  =  5ax*z"dx  H  --     ---  . 


Ex.  9.  Differentiate  u  =  x*y*.  Result,  'LL, 

2^V 

?) 

Ex.  10.  From  i/2  ==  2px  find  the  value  of  dy.  dy  =  -dx. 

Ex.  11.  From  A*\f~  +  ^=a;2  =  A^E^  find  the  value  of  dy. 

B*x. 

d  y  =  --  T—  fe 
^^ 

Ex.  12.  From  A*y*  —  B*x*  ==  —  A*B*  find  the  value  of  dy. 

B*x. 

dy  =  --  dx. 

* 


Ex.  13.  From  #2  +  1/2  =  ^2  nnd  the  value  of  dy.      di/  =  —  -dx. 
Ex.  14.  From  2xy2  —  ay2  =  ^73  find  the  value  of  dy. 


18  THE  DIFFERENTIAL  CALCULUS. 


T^       -rr     TVJT  r>        i,    j 

Ex.  15.  Differentiate  u  =  —  .        Result,  du 


1 

Ex.  1C.  Differentiate  y  =  -. 

a; 


Ex.  17.  Differentiate  «  =  -  - 


-~  . 

Ex.  18.  Differentiate  y  =  -—  . 

5 


SUG. — Do  not  treat  this  as  a  fraction  under  Rule  7. 
Ex,  19.  Differentiate  u  = 
Ex.  20.  Differentiate  u  = 


. 
dij  —  --  x  2xd#  =  -—do:. 


OPEBATION.       du  =  -  +    ««)   - 


a?)  —  (4da  +  2gdr)(2a^  —  3) 


Ox 


(4x  -j-  a*)* 

SUG'S.  —  The  first  step  is  the  application  of  the  rale  for  fractions,  since  the  func 
tion  is  a  fraction  with  a  variable  numerator  and  a  variable  denominator.  The 
second  step  is  to  perform  the  differentiation  of  2x*  —  3,  and  4z  -f-  x2.  This  step 
involves  the  rules  for  constant  factors,  variables  affected  with  exponents,  constant 
terms,  and  the  sum  of  variables.  The  remainder  of  the  work  is  reduction  and 
addition  of  terms. 


.       _      . 

Ex.  21.  Differentiate  u  =  --  .  du  =  —  -  -  —  ~  dx. 

&  —  jr2  (a*  —  x^y 

Ex.  22.  Differentiate  y  =  a~  '*-.  dii  •=*  —  -dx. 

x  x* 

Ex.  23.  Differentiate  y  =  iif-.  dy         1  ~  2'r 


. 
1  -f  ^  (1 


Ex.  24.  Differentiate  y  =  ;  —  —  • 
y        1  —  -  x* 


Ex.  25.  Differentiate  y  =  - — — . 
Ex.  26.  Differentiate  v  =  3.rw  —  4. 


Ex.  27.  Differentiate  y  =  2m#».  ^y  = A- 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS.  19 

Ex.  28.  Differentiate  u  =  2iui»y\ 

du  =  %mx  »  ydx  -f  2o;  "?/  "  dy* 

Ex.  29,  Differentiate  y  =  —.  dy  =  -     —  ^- 

J7  J7 

Ex.  30.  Differentiate  y  =  v/^3  —  aa. 

OPERATION,—  By  tlie  special  rule  for  the  square  root   (57),   we  have  <Jy  = 


Ex.  31.  Differentiate     = 


or 


/*-_  7* 

Ex.  32.  Differentiate  g  =  avx  —  -. 

i) 


Ex.  33.  Differentiate  #  =  v/ao;  +  bx*  + 


Ex,  34  Differentiate  y  —  (ow:*  — 

SOLUTION.  —  Begarding  aa2  —  a3  as  a  variable,  it  is  affected  wiih  the  exponent  4  ; 
hence  we  have  dy  =  4(oa;2  —  -x3)3  X  d(axz  —  x3),  the  operation  of  differentiating 
the  variable  ax*  —  a;3  being  as  yet  unperformed.  Performing  this  operation  and 
reducing,  we  bave«dijf  —  4(o#3  —  a3)3  X  (2«aj  — 


Ex,  35.  Differentiate  y  =±  (a 
Ex.  36,  Differentiate  y  =  («»  + 

a  . 

Ex,  37.  Differentiae  y  =         ~  ^  =     - 


Ex.  38.  Differentiate  y  «  (1  -f  2««)(1  -f 

SOLUTION,  —  Begarding  this  function's  the  product  of  the  two  variables  1  -f-  2x2 
and  1  +  4x3,  we  have  dy  =  d(l  +  2x2)  X  (1  -f  4«3)  -f  ^C1  +  4^)  X  I1  +  2ar')- 
Performing  the  operation  of  differentiating  1  -f-  2^'2  and  1  -f  ^x3-,  we  have  dy  =s 
-f  12x2(1  -f  2«a)<2a;^s  4^(1  4  3x  +  10x:3)dx. 


Ex,  3&  Differentiate  y  =*=  (a?  4  a)(3.r2  4 

Ex.  40.  Differentiate  y  ~ 
Ex.  41.  Differentiate  v  = 


^i/i= 

(15^4  +  35j?2  4  (MP/CMR, 

Xs 

(1  -f  *)«* 
a 
(«—'»)•" 

3^2  4  .^3rf 

"a-f  ^)3  a 

3a^r 

^  =  ^rr7^ 

20  THE   DIFFERENTIAL   CALCULUS. 


ax* 
Ex.  42.  Differentiate  y  =  —  ;  --  -.  dy= 


.  . 

(ab  —  x*y  (ab  —  x*)* 

Ex.  43.  Differentiate,  without  first  expanding,  y  =  (1  -j-  .r)4(l  -f  #2)2. 
<fy  =  4(1  +  ^)3(1  +  #2)(1  +  x  4-  2#*)<te. 

Ex.  44.  Differentiate  y  —  a;*  —  v/1  —  #3. 

' 

4 


Ex.  45.  Differentiate  M  =  \/%ax  —  x*.  du  =  x)_^_ 


Ex.  4G  .  Differentiate  u  =  Va*  +  a;1'  x  Vb* 


+  x*  X  v6*  -f 


/v» 

Ex.  47.  Differentiate  y  =  — — ' -.  dy  = 

S/0*  —  ar»  v7  (  a«  —  x* ) : 

Ex.  48.  Differentiate  y  = 


SUG'S.  ' 


\/l  +  X* 


.-f  ,          dx(\  -f-  x2)  —  x"dx  dx 

fl?(l  +  x2)     dx  =  —  -  ---  =  -     —  -.     Or,  we  may  apply  the  rules 

(If**)  (l+ar2)* 

t?x\/l  4-  a;14  —  .rr7\/l  +  «2    • 
for  a  fraction    and  a  square    root,  thus    dy  = 


dx\/l  -4-  x*  —  x- 


1+& 

xdx 


+  ^        (?^(1  -f-  x'2)  —  tfdx  dx 


(1  +  a?)*  (1  +  x*)* 


Ex.  49.  Differentiate  w  =  (1  +  x)     T~v.          du  = 


Ex.  50.  Differentiate  u  =  —        .  du  = '— . 

/<!-*•)•  (l_*)t 

Ex.  51.   Differentiate  «  = 


—  a;4 


Ex.  52.  Differentiate  u  =  V'./;  4-  \/l  4-  ^2. 

SUG'S.— Squaring    u2   —  x  4-  \/l  4-  x'-1.      2t«Zu  =  dx  4-        X  X — .      du   = 

v/1  4-  ^ 


DIFFEEENTIATION   OF  ALGEBRAIC   FUNCTIONS.  21 


xdx 

(-T    4.    v/i    4.   x<)dx 

**   '    v/i  4-  ** 

v/l  4-  £ 

(.r  4-  v/i  -f  x-)tiu 

2  a 

2Vx4-  v/14-x2             2  v/l  4-  x^Vx  4-  v/l  4- 
-.      Or,   we  may  differentiate  without   squaring,   thus 

x" 
du, 

Vx  4    v'l  +  ;e--Yfa 

2v/l  4-  x^ 

^     (          xdx 

V/l   +  X2 

(X  _|_  v/i  4.  a;2)(7.r 

V  x  4-  v/l  4-  ir.--(te 

v/I 


Ex.  53.  Differentiate  u  = 


V  a1  +  x*  •  —  x 


SUG'S.—  As  the  denominator  is  more  involved  in  the  differential  of  a  fraction 
than  the  numerator,  it  is  expedient  to  reduce  the  fraction  to  a  form  having  as  sim 
ple  a  denominator  as  possible,  nationalizing  this  denominator,  we  have  u  = 

I  a*     da  =  <2£±^  4. 


Ex.  54.  Differentiate  w  = 


*  +  1  4-  x 

*•-». 


Ex.  55.  Differentiate  u  =  \j  —      — 7=. 

1  +  Vx 


__v/.c        Vl  —  v/x        v/l  — x       ,  da; 

SUG  S.       M  = 


v/x 


Ex.  5G.   Differentiate  w  = 


Ex.  57.   Differentiate  w  = 


Ex.  58.  Differentiate  u  =  v^/^;  -  V  Vx  -f  1. 


=  — - — dx. 


22 


THE  DIFFEKENTIAL  CALCULUS. 


Ex.  59.  Differentiate  u  =  \2x — 1 — V  2x — l  —  Vzx — 1 — ,  etc., 
to  infinity. 


SUG'S.— We  have  u  =  V'2x  —  1  —  u  ;    whence  w2  =  2ar —  1  —  u>  and  u 


Vtix 


Ex.  60.  Differentiate  u  = 


iate  u  =  /I  fa  --  = 
\L         v/: 


du  = 


+  ^  (<.,_*»)* 

V  X 


ILLUSTRATIVE    EXAMPLES. 

[Nora.— The  following  examples  are  designed  to  illustrate  more  fully  the  significance  of  the- 
process  of  differentiation.] 

Ex.  1.  In  a  parabola  whose  parameter  is  12,  which  is  increasing 
the  faster  at  x  =  2,  the  ordinate  or  the  abscissa,  and  how  much  ?  At 
#  =  3?  At  a;  =  8  ?  At  #=24?  How  does  the  relative  rate  of 
change  vary  as  we  recede  from  the  vertex  ?  At  what  point  are  ordi 
nate  and  abscissa  varying  equally  ? 

SOLUTION. — The  equation  of  this  parabola  is  y-  —  12o?. 
Differentiating,  we  have  dy  =  -dx.  Now,  as  differentiat 
ing  is  the  process  of  finding  the  difference  between 
two  consecutive  states  of  a  function,  dx  represents  one 
of  the  infinitesimal  increments  of  x,  as  D  D  ,  and  dy 
the  contemporaneous t  infinitesimal  increment  of  y,  as 


P'E.     We,  therefore,  learn  from  dy 


-dx  that  in  gen- 

y 


eral  dy  is  -  times  as  great  as  dx  ;  or,  in  other  words,  that 


FIG.  11. 

y  changes  "  times  us  fast  as  x.     At  P  where  x  =  2,  y  =  >/^4.     Hence,  at  this 
point,  dy  =  -^cfa  =  ix/Cdx  ;  that  is,  7;  is  increasing  nearly  l£  times  as  fast  as  x. 


At  P"  where  x  —  3,  y  =  6,  and  dy  =  dx  ;  that  is,  x  and  y  are  increasing  equally. 
In  general,  at  the  focus  the  ordinate  and  abscissa  of  a  parabola  are  increasing 
equally,  since  at  this  point  y  =  p.  At  PIV  where  x  =  8,  y  is  increasing  only  about 
.6  as  fast  as  x.  At  PVI  where  x  =  24,  y  is  increasing  at  the  still  slower  rate  of 
about  .35  as  fast  as  x.  Finally,  it  is  evident,  from  a  slight  inspection  of  the  figure. 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS.  23 

that  y  increases  less  and  less  rapidly  as  x  becomes  larger,  a;  continuing  to  increase 
at  a  uniform  rate.  At  x  =  oo,  y  ceases  to  increase,  L  e.  the  branches  become  par 
allel  to  the  axis  of  x. 

Ex.  2.  Examine  the  relative  rates  of  change  of  the  ordinate  and 
abscissa  in  the  ellipse. 

SOI.CTION.  —  Differentiating   A°y*  •}-  B"x*  =  AZB\  we  find  <Jy  =  —     -^-dx  = 
dx.     On  this  we  observe  1st,  That  the  —  sign  shows  that  x  and  v 


—  x* 

are  decreasing  functions  of  each  other  ;  that  is,  that  as  x  takes  an  increment  y 
takes  a  decrement.     This  is  evident  from  a  consideration  of  the  curve.     2nd,  That 

in  general  terms  y  diminishes  -  -  ----  times  as  fast  as  x  increases.     3rd,  At 

A\/A*  ~  B» 
x  =  0,  *.  «.  at  th«  extremity  of  the  conjugate  axis,  y  is  not  increasing  or  decreas 

ing,  since  liere  -  -•  =  0,  and  dy  =  0  -  dx  =  0.     At  the  extremity  of  the 


transverse  axis  dy  =  —  oc  -  dx,  Le.  y  is  decreasing  infinitely  faster  than  re  increases. 
There  are,  therefore,  all  relative  rates  of  change  between  x  and  y  from  0  to  oc. 
Moreover  as  x  begins  to  increase  from  0,  y  commences  to  decrease  (at  first  slowly, 

as  the  fraction  -  :  is  small  when  x  is  small),  and  then  more  and  more 

AVA*  —  x* 
rapidly  as  x  increases,  till  it  reaches  an  infinitely  rapid  rate  of  decrease  at  x  —  A. 

This,  it  is  easy  to  see,  is  the  law  of  change  in  the  fraction  -  ~       _   as  x  in- 

—  x* 


creases.  The  same  law  is  also  rendered  probable  from  an  inspection  of  the  curve. 
Finally,  we  may  inquire  at  what  point  the  relative  rates  of  change  sustain  any 
given  relation  to  each  other,  as,  for  example,  when  y  decreases  twice  as  fast  as  x  in 
creases,  or  just  as  fast,  or  10  times  as  fast.  Thus  when^  decreases  twice  as  fast  as 

Bx 

x  increases,  we  must  have  dy  •=  —  2<fx,  L  e.  -  —  --  -  =  2L     From  this  we  find 


;  lience  at  these  points,  y  is  diminishing  twice  as  fast  as  x  is 


increasing. 


Ex.  3.  A  boy  is  running  on  a  horizontal  plane  directly  toward  the 
foot  of  a  tower  100  feet  in  height.  How  much  faster  is  he  nearing  the 
foot  than  the  top  of  the  tower  ?  How  far  is  he  from  the  foot  of  the 
tower  when  he  is  approaching  the  base  twice  as  fast  as  he  approaches 
the  top  ?  How  far  off  must  he  be  to  be  approaching  both  base  and 
top  equally  ?  Where  is  he  when  he  is  not  approaching  the  top  at  all, 
or  is  making  infinitely  more  progress  toward  the  base  than  towards 
the  top  ?  "When  he  is  at  200  feet  from  the  base  of  the  tower  how 
much  faster  is  he  approaching  the  base  than  the  top? 

STTG'S.  —  Let  AB  represent  the  tower,  and  AX  the  line  in  the  plane  of  the  base 
in  which  the  boy  is  approaching  the  base.  Suppose  the  boy  at  any  point,  as  P 


24  TEE  DIFFERENTIAL   CALCULUS. 

and  let  AP  =  x,  and  PB  =  y.      Then 
2/2  —  x°~  =  10000.    Whence  dy  =  -dx.    Hence 

we  see  that  in  general  he  is  only  approach- 

fl£ 
ing  the  top  an  -th  part  as  fast  as  he  is  the 

base  ;  i.  e.,  letting  PP'   represent  an  infin 
itesimal  element  of  the  distance  to  the  foot  of  the  tower,  P  F  represents  a  contem 
poraneous,  infinitesimal  element  of  the  distance  to  the  top  ;  and  also,  that  P  F    is 

an  -th  part  of  PP'.     Secondly,  when  he  is  approaching  the  foot  of  the  tower 
y 

1  x      1 

twice  as  fast  as  he  is  the  top  ;  we  have  dy  =  ^dxy  or  -  =  -  ;  whence  y  =  2x.     But 

100 

y2  —  x2  =  10000  ;  and,  substituting,  3^  =  10000,  or  x  =  -—  =  58  nearly.     Lastly, 

\/3 

when  he  is  at  200  feet  from  the  base  y  =  v/SOOOO  =  224  nearly,  and  -  =  -—  =  — 

y       224       28 

or  or 

nearly.     Hence  dy  =.  ^-dx,  or  he  is  approaching  the  top  --  as  fast  as  he  is  the 

2o  2o 

base.     [Let  the  pupil  decide  the  other  points  himself.] 

Ex.  4.  A  ship  is  sailing  northwest  at  15  miles  an  hour.     At  what 
rate  is  she  making  north  latitude  ? 

Ans.t  At  10.G05+  miles  an  hour. 

Suo's. — Let  y  represent  any  distance  run  in  the  northwest  course,  and  x  the  cor 
responding  northing.  Then  as  the  course  is  northwest  there  is  made  in  the  same 

2.r 
time  x  westing,  and  we  have  y-  =  2x2.     From  this  dy  =  —dx,  and  the  ship  is  run- 

2x  —  2r 

ning  —  times  as  fast  as  she  is  making  northing.     But  y  —  x\/2,  whence  -—  = 

2x  -  - 

'—=.  =  v/2,  and  dy  =  \/l2dx,  or  dx  =  iv/2c7?/ ;   L  e.,  she  is  making  northing 
xv  2 

.707-}-  as  fast  as  she  is  running. 

Ex.  5.  In  the  function  y  ==  27x  -f-  3^2,  required  the  value  of  x  when 
y  is  increasing  45  times  as  fast  as  x.  Result,  x  =  3. 

Ex.  6.  What  is  the  relative  rate  of  variation  of  the  side  and  alti 
tude  of  an  equilateral  triangle  ?  i.  e.,  if  the  side  takes  an  infinitesi 
mal  increment,  what  is  the  contemporaneous  infinitesimal  increment 
of  the  altitude  ?  When  the  side  is  increasing  at  the  rate  of  2  inches 
per  second  how  rapidly  is  the  altitude  increasing  ?  Is  the  relative  rate 
of  increase  constant  or  variable  ;  that  is,  does  the  altitude  increase 
more  or  less  rapidly  in  comparison  with  the  side  when  the  side  is 
small  than  it  does  when  it  is  large,  or  is  the  relative  rate  of  increase 
always  the  same? 

Suo's.— Let  y  =  the  altitude  and  x  one  of  the  sides  of  the  triangle.     Then 


LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS.         25 

3x  3x 

2/2  =  Ix2,  and  dy  =  --dx  =  — —dx  =  ^\/'3dx.     Hence  we  see  that  the  infinitesi- 
ty  2v/3x 

mal  increment  of  y  is  always  iv/3  times  as  much  as  the  contemporaneous  infini 
tesimal  increment  of  x.  When  x  is  increasing  at  2  inches  per  second  y  is  increas 
ing  2\/3  times  2  inches,  or  \/3  inches  per  second. 

Sen. — The  student  should  now  be  able  to  comprehend  with  considerable 
clearness  the  object  of  the  Differential  Calculus;  viz.,  having  given  the 
relation  between  finite  values  of  variables,  to  find  the  relation  between  the 
contemporaneous  infinitesimal  increments  of  those  variables,  or  their  rela 
tive  rate  of  change.  Thus,  in  the  last  example,  the  relation  between  the 
altitude  and  one  side  of  an  equilateral  triangle,  y2  =  }.»*,  is  the  relation 
between  finite  values  of  the  variables,  from  which  we  find  the  relation  be 
tween  the  contemporaneous  infinitesimal  increments  dy  and  dx>  by  the  Dif 
ferential  Calculus. 


SECTION  II. 
Differentiation  of  Logarithmic  and  Exponential  Functions,* 

58.  The  Modulus  of  a  system  of  logarithms  is  a  constant 
factor  which  depends  upon  the  base  of  the  system  and  characterizes 
the  system. 

50.  3*TOp. — The  differential  of  the  logarithm  of  a  variable  is  the 
differential  of  the  variable  multiplied  by  the  modulus  of  the  si/stem,  di 
vided  by  the  variable ;  or,  in  the  Napierian  system  the  modulus  being  1, 
the  differential  of  the  logarithm  is  the  differential  of  the  variable  divided 
by  the  variable. 

DEM. — Let  y  =  x",  n  being  constant.      Then  log  y  =  n  log  x.     Differentiating 

dy 

•u  =  x",  we  have  dii  =  nxn—ldx,  or  n  = : —  =  -^—  =  --,  since  x"""1  =  -.     Again, 

x"-ldx      y          dx  x 

~c/x .     — 

X  X 

whatever  the  differentials  of  logy  and  logx  are,  we  have  d(logy)  =  n  •  d(logx\ 

or   n  =  -J — .      Placing   these  values  of   n  equal   to   each  other,   we   obtain 

dj/° 

-— -— -—  =  — .     Now  let  m  be  the  factor  by  which  --  must  be  multiplied  to  make 
a  (log  x)       dx  y 


it  equal  to  <Z(logy),  then  is  d(Ioga)  =  ~~. 


*  See  32,  33. 


26  THE  DIFFERENTIAL  CALCULUS. 

We  are  now  to  show  that  m  is  a  constant  depending  upon  the  base  of  the  system. 

dy 


To  do  this  take  y  =  zn>,  from  which  we  find  as  before  ri  =  f  '  °^  =  ~  .     But  m 

z)   dz 


is  the  ratio  of  d(logy)  to  —  ;  hence  d(logz)  =  -  .     Thus  we  see  that  in  any  case 

the  same  ratio  exists  between  the  differential  of  the  log.  of  a  number,  and  the  differ 
ential  of  the  number  divided  by  the  number.  Therefore  m  is  a  constant  factor. 
To  show  that  m  depends  upon  the  base  of  the  system  we  have  but  to  recur  to  the 
definition  of  a  logarithm  to  see  that  the  only  quantities  involved  are  the  number,  its 
logarithm,  and  the  base  of  the  system.  Of  these  the  two  former  are  variable,  whence, 
as  the  base  is  the  only  constant  in  the  scheme,  m  is  a  function  of  the  base.* 

Finally,  as  m  depends  upon  the  base  of  the  system,  the  base  may  be  so  taken 
that  m  =  1.  The  system  of  logarithms  founded  on  this  base  is  called  the  Napie 
rian  system.  Q.  E.  D. 


00.  jProf).  —  The  differential  of  an  exponential  function  with  a 
constant  base  is  the  function  itself  ',  into  the  logarithm  of  the  base,  into  the 
differential  of  the  exponent,  divided  by  the  modulus. 

DEM.  —  Let  y  =  ax.     Taking  the  logarithms  of  both  members  logt/  =  xloga. 


_..„  mdy 

Differentiating  —  -  =  log  adx,  or  dy  =  --  ,  remembering  that  y  =  a*,  and 

that  log  a  is  constant.     Q.  E.  D. 

01.  COR.  1.  —  TJie  differential  of  an  exponential  function  with  a  con 
stant  base,  taken  with  reference  to  the  Napierian  system,  is  the  function 
itself,  into  the  logarithm  of  the  base,  into  the  differential  of  the  exponent. 
Thus  if  y  ==  ax,  dy  =  ax  log  adx. 

02.  COR.  2.  —  If  the  base  of  the  exponential  is  the  base  of  the  system  of 
logarithms  in  reference  to  which  the  differentiation  is  made,  we  have,  in 

general,  dy  =  -  —  -  ,  or  in  the  Napierian  system  dy  =  ezdx,  since  the  log 

arithm  of  the  base  of  a  system,  taken  in  that  system,  is  1,  and  in  the  Na- 
pierian  system  e  is  used  to  represent  the  base  and  m  =  1. 


G3m  IP  Top.  —  The  differential  of  an  exponential  with  a  variable  base 
is  best  obtained  by  passing  to  logarithms,  and  then  differentiating. 

ILL.—  Let  u  =  yx.    Passing  to  logarithms,  log  u  =  x  log  y.     Differentiating,  we 

,  mdu  mxdti  it  log?/  dx    .    u  x  dy 

have  in  general,  —  —  =  logi/dx  4-   —  —  .  whence   du  =  -  -  ---  --  -  = 

u  y  my 

yr  \ngydx        tf'xdy 

-  -j-  :  -  '-.     If  the  logarithms  are  taken  in  the  Napierian  system,  TO  =  1, 


*  What  this  relation  is,  it  does  not  concern  us  at  present  to  know.    It  will  be  determined  here 
after. 


LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS.         27 

and  du  —  yx  log  y  dx  -j-  y*~lxdy.     If  in  addition  y  =  x,  so  that  it  =  se*,  dtt  = 
jc*(logx  -f-  l)cto. 


EXERCISES. 

[NOTE. — The  following  exercises  are  designed  to  familiarize  the  rules  for  differentiating  loga 
rithmic  and  exponential  functions,  and  give  the  needed  facility  in  applying  them.] 

Ex.  1.  Differentiate  u  —  x  log  x. 

du  =  log-refo  -f  wiefar,  or  (log a;  +  I)dx. 

Ex.  2.  Differentiate  u  =  log  a:2.  du  =  2m--,  or . 

iJC  SG 

Ex.  3.  Differentiate  u  =  log2  #.        du  =  2m  log  #— -,  or  2  log  # — . 

x  x 


Ex.  4.  Differentiate  u  —  x- 
du 


(  l) 

=  of  'x'  \  log  tf  (log  x  -f  1)  +  -  \  dx.* 

-     r •        ...    v  •w  ' 


Ex.  5.   Differentiate  M  =  aloga;.  du  == 


Ex.  6.  Differentiate  w  =  log  v  1  —  ara.  du  =  — 

Ex.  7.  Differentiate  u  =  log  (3^2  -f  #).  c?u  =  -        —do?. 

Ot/7      ~y~     *37 

Ex,  8.  Differentiate  u  =  log  (or  +  v/1  +  a?*).         du  =  - 


v/1 

Ex.  9.  What  is  the  differential  of  w  =  ax  in  the  common  system 

2    2 
when  a  is  the  base  of  the  system  ?  du  =  —  a1  awfcp. 

Ex.  10.  Differentiate  w  =  elog*,  in  the  common  system,  e  being  the 

wrfar .  mudx 

base  of  the  Napierian  system.  du  = log  e  = . 

x  x 

SUG. — If  the  student  has  studied  the  subject  of  logarithms  as  usually  presented 
in  our  higher  Algebras,  he  has  learned  that  the  common  logarithm  of  the  Napier 
ian  base  is  the  modulus  of  the  common  system  ;  i.  e. ,  in  this  example  log  e  =  m. 
This  fact  will  also  appear  hereafter. 


VX'2  +  1  —  x 
Ex.  11.  Differentiate  u  =  log 


gUG. — First     rationalize    the    denominator    of   the    fraction,    obtaining    u  = 

log  (v/x'-f-l  —  a;)2  =  2  log(  v/£2-t-l  —  »),  and  then  differentiate,     du  = '— . 

*  The  student  will  observe  for  himself  whether  common  or  Napierian  logarithms  are  used. 


28  THE  DIFFEKENTIAL  CALCULUS. 

64.  Sen.  — The  differentiation  of  algebraic  functions  is  often  performed 
with  greater  facility  by  first  passing  to  logarithms. 

Ex.  12.  Differentiate  u  =      t-. 


SUG.  —Passing  to  logarithms  we  have  log  u  =  log  (1  -f-  #2)  —  l°g  (1  —  #2)-     Differ- 

du        2xdx       —  2xd.c  4.rtZ.r 

entiatmg,    —  = = .     .  • .  du  = 

u        1-f-^-        1  —  &'2          (l-t-#")(l — x'2) 

— X  — — —  = •     This  example  illustrates  the  method 

(14-x*)(l  —  x*)       1—  x2       (1— a?*)4 

referred  to  in  the  scholium,  although  the  student  will  find  the  direct  method  quite 
as  expeditious. 

Ex.  13.  Differentiate  u  =  x(a*  -f-  &)va-  —  x'2,  by  first  passing  to 


a4  -f-  a-x2  —  4:X4 

logarithms.  da  = —        - — dx. 

va2  —  j;2 

Ex.  14.  Differentiate  u=  (a*  +  I)2.        da  =  2ax(ax  -f  1)  log 

Ex.  15.  Differentiate  u  =       ~~  ,.  du  =  -^- 

a?  -f  1  (az+  I)2 

G5.  CoR.—TJie   ordinary   rule    (SO)  for   differentiating  a  variable 
affected  with  an  exponent  applies  when  the  exponent  is  imaginary. 

DEM  — Let  u  =  xa  *~l.    Passing  to  logarithms,  log  u  =  a  \/ —  1  log  x.     Differentiat- 

du            , — -dx                           . — -udx  , — -   o"Vi^_i , 

ing,  —  =  av—  1— .     . • .  du  —  av —  1 =  av —  1  x  dx.     Q.  E.  r>. 


ILLUSTRATIVE    EXAMPLES. 

Ex.  1.  Which  increases  the  faster,  a  number  or  its  logarithm  ? 

SOLUTION.  — Let  x  represent  any  number  and  y  its  logarithm,  so  that  y  =  log  x. 
We  now  wish  to  find  the  relation  between  the  contemporaneous,  infinitesimal  in 
crements  of  x  and  y  ;  i.  e.,  if  the  number  (x)  changes  how  does  the  logarithm  (y) 

change?    Hence  we  differentiate,  and  have  dy  =  —dx.     From  this  we  see  that  the 

increment  of  the  logarithm  (dy}  is  —  times  the  increment  of  the  number  (cZx). 

Therefore  when  x  <^m  the  logarithm  increases  faster  than  the  number  ;  when 
x  >  m  the  logarithm  increases  slower  than  the  number  ;  and  when  x  =  m  they 
increase  equally. 

[NOTE.  —The  student  should  not  fail  to  see  in  every  such  example  the  real  object  of  the  Dif 
ferential  Calculus  (42).  In  the  last  example  the  relation  between  finite  values  of  the  variables  x 
and  y  is  y  =  log.  x.  The  relation  between  the  contemporaneous,  infinitesimal  elements  of  these 
variables  is  found  by  differentiating,  this  being  the  object  of  the  Differential  Calculus.] 

Ex.  2.  When  the  number  is  2124  and  is  conceived  as  passing  on  to 
larger  values  by  the  law  of  growth  of  continuous  number,  i.  e.  by 


LOGARITHMIC  AND   EXPONENTIAL  FUNCTIONS.  29 

taking  on  infinitesimal  increments,  how  much  faster  is  the  number 
increasing  than  its  common  logarithm  ?  If  this  relative  rate  of  change 
continued  uniform  (which  it  does  not)  while  the  number  passed  to 
2125,  i.  e.  increased  by  1,  how  much  would  the  logarithm  have  in 
creased  ? 

SOLUTION. — Letting  x  be  any  number  and  y  its  logarithm,  we  have  found  that 
dy  =  —  dx.     But  m,  the  modulus  of  the  common  system  =  .'43429448.     Hence 

when  a;  =  2124,  we  have  dy  =  '—  -^ dx  =  .000204dt,  or  the  increment  of  the 

logarithm  is  .000204  part  of  the  increment  of  the  number.     The  number  is,  there- 

-j  nnnnnr) 
fore  increasing  -        — ,  or  about  4902  times  as  fast  as  the  logarithm.     Secondly, 

«v9c 

If  this  relative  rate  of  change  continued  the  same  while  the  number  passed  from 
2124  to  2125,  I  e.  increased  by  1,  the  logarithm 
would     increase     once    .000204,    or    .000204. 
Hence  the  logarithm  of  2125  would  be  .000204 
larger  than  the  logarithm  of  2124. 

GEOMETRICAL  ILLUSTRATION. — Let    M  N   be 


A/ 

the  curve  whose  equation  is  y  =  log  x.     Take         M 

AD  =  2124  ;  then  will  P  D  represent  its  loga 
rithm.      Let    DD'   represent    dx;    then  will  j,       -.0 
P'  E  represent  dy.* 

Ex.  3.  The  common  logarithm  of  327  is  2.514548.  "What  is  the  log 
arithm  of  327.12,  on  the  hypothesis  that  the  relative  rate  of  change 
of  the  number  and  its  logarithm  continues  uniformly  the  same  from 
327  to  327.12  that  it  is  at  327? 

SUG.— At  327  dy  =     '    "  ~  '  -dx  =  .001328<te.     Now  as  the  number  327  increases 

•  >'- 1 

.12  to  become  327.12  ;  according  to  the  hypothesis  the  logarithm  increases  .12 
times  .001328  or  .000159.     Hence  the  logarithm  of  327.12  is  2.514707. 

SCH. — The  hypothesis  that  the  relative  rate  of  change  of  a  number  and 
its  logarithm  continues  constant  for  comparatively  small  changes  in  the  num 
ber,  is  sufficiently  accurate  for  practical  purposes,  and  is  the  assumption  made 
in  using  the  tabular  difference  in  the  table  of  logarithms,  as  explained  in  THE 
COMPLETE  SCHOOL  ALGEBKA  (125],  and  in  the  introduction  to  the  table  of 
logarithms  (14)  in  the  volume  on  Geometry  and  Trigonometry. 

Ex.  4.  What  should  be  the  tabular  difference  in  the  table  of  loga 
rithms  for  numbers  between  2688  and  2689?  Ans.,  .00016156+. 

QUERY.— Hqw  is  it  that  the  tabular  difference  found  in  the  table  of  logarithms  for 

*  The  figure  is  necessarily  out  of  proportion,  as  the  true  relation  of  y  and  x  requires  that  A  D 
be  nearly  700  times  as  long  as  P  D . 


30  THE  DIFFERENTIAL  CALCULUS. 

numbers  between  2688  and  2689,  is  161  ?  Show  bow  the  method  of  rising  this 
tabular  difference  makes  the  result  agree  substantially  with  the  method  of  inter 
polating  now  being  presented. 

Ex.  5,  According  to  the  arrangement  of  our  common  tables,  show 
that  the  tabular  difference  corresponding  to  7487  is  58. 


SECTION  III. 
Differentiation  of  Trigonometrical  and  Circular  Functions 

TBIGONOMETRICAL    FUNCTIONS. 

00.  Prop. — The  differential  of  the  sine  of  an  arc  (or  angle)  is  the 
cosine  of  the  same  arc  into  the  differential  of  the  arc. 

DEM, — Let  a;  be  any  arc  (or  angle)  and  y  its  sine,  f.  e.  let  y  =  sin/c.     If  x  takes 
an  infinitesimal  increment  (dx),  let  dy  represent  the  contemporaneous  infinitesi 
mal  increment  of  y.     Then  the  consecutive  state  of  the  function  is 
y  -f-  dy  —  sin  (x  -f-  dx)  =  sin  x  cos  dx  -f-  sin  dx  cos  x. 

Now  cos  dec  —  1,  since  as  an  angle  grows  less  its  cosine  approaches  the  radius  in 
value,  and  at  the  limit,  is  radius.  Moreover,  as  an  angle  grows  less  the  sine  and  the 
corresponding  arc  approach  equality,  and  at  the  limit  we  have  sin  dx  =  dx*  The 
consecutive  state  may  therefore  be  written  y  -f-  dy  —  sin  x  -f-  cos  a-  dx. 

From  this  subtract  ?/  =  sin  x 

and  -we  have  dy  =  cos  x  cLc, 

•which,  being  the  difference  between  two  consecutive  states  of  the  function,  is  the 
differential.  Q.  E,  D. 

67.  Prop. — The  differential  of  the  cosine  of  an  arc  (or  angle]  has 
the  opposite  algebraic  sign  from  the  function,  and  is  numerically  equal  to 
the  sine  of  the  same  arc  into  the  differential  of  the  arc. 

DEM. — Let  x  be  any  arc  (or  angle)  and  y  its  cosine,  so  that  y  =  cos  a;.  Since 
cos  x  =  sin  (90°  —  x)  we  have  y  =  sin  (90°  —  jc).  Differentiating  this  by  the  pre 
ceding  proposition,  we  obtain 

dy  =  cos  (90°  —  x)  X  d(90°  —  x)  —  cos  (90°  —  x)(—  dx)  —  —  sinxdx, 
since  dv90°  —  x)  —  —  dx,  and  cos  (90°  —  x)  =  sin  x.     Q.  E.  D. 

€8.  Sen.— The  opposition  in  the  signs  of  the  differential  of  the  cosine, 
and  of  the  corresponding  arc,  signifies  that  they  are  decreasing  functions 
of  each  other  (40}  ;  f .  e. ,  if  one  takes  an  increment  the  other  suffers  a  de 
crement. 

*  The  student  may  be  inclined  to  say  that  at  the  limit  eiuda;  *=  0.  This  is  true,  and  no  error 
would  follow  from  the  assumption  ;  but  the  statement  in  the  demonstration  is  equally  true,  and 
we  consider  sin  dx  =  dx  instead  of  =  0,  simply  because  we  do  not  wish  to  have  dx  vanish  from 
the  formula,  our  object  being  to  find  the  relation  between  dy  and  dx. 


TRIGONOMETRICAL  FUNCTIONS.  31 

00  .  JProp*  —  The  differential  of  the  tangent  of  an  arc  (or  angle)  is 
the  square  of  the  secant  of  the  same  arc  into  the  differential  of  the  arc  ; 
or  for  the  square  of  the  secant  we  may  write  the  reciprocal  of  the  square 
of  the  cosine. 

DEM.  —  Let  v  =  tanx.     Now  tan  x  =  -  -,  whence  y  =  -  .     Differentiating 

cos  x  cos  x 

this,  observing  that  -  '—  is  a  fraction  with  a  variable  numerator  and  denominator, 
cos  x 

and  hence  can  be  differentiated  by  the  rule  for  fractions  (54:},  and  the  two  propo- 

cns  x  d(  Kin  x]  —  sin  x  d  cos  x)       cos  -  x  dx  4-  sin-  x  dx 
Bitions  (06,  6  7],  we  have  dy  = 


cos2  x  4-  sin2  x  ,  1 

5=5  ---  !  -  dx  ==  -  dx  =  sec2  x  dx.     Q.  E.  D. 
cos-  x  cos*  x 

ANOTHER  DEMONSTBATION.  —  The  conseciitive  state  of  the  function  y  =  tan  x, 

tan  x  -4-  tan  dx  tan  x  -f-  dx 

being  y  4-  dy  =  tan  (x  4-  dx}  =  -  -  •  --  -  =  -  -  ~  ---  --,  the  difference 

1  —  tauxttuidc         1  —  tanxdc 

between  the  two  states,  i.  e.   the  differential  is  dy  —     a    •    ~t~  (  •_,  —  ^an  x  __ 

1  —  t&uxdx 

-   ~*~  tan2a;.-dx  =  (1  +  tan2x)dc  =  sec2  x  dr.     [Let  the  student  give  the  detailed 
1  —  tau  x  dx 

explanation  of  the  process.] 


70.  Prop* — The  differential  of  the  cotangent  of  an  arc  (or  angle) 
has  the  opposite  algebraic  sign  from  the  function,  and  is  numerically  equal 
to  the  square  of  the  cosecant  of  the  same  arc  into  the  differential  of  the 
arc ;  or  for  the  square  of  the  cosecant  we  may  ivrite  the  reciprocal  of  the 
square  of  the  sine. 

DEM.— Let  y  =  cot  x  =  tan  (90°  —  x).     Differentiating  by  the  last  proposition 

1 
dy  =  sec2  (90°  —  x)  X  d(90°  —  x)  —  cosec2  x(-^-  dx)  =  —  cosec2  xdx,  or r—p- dx. 

Q.  E.  D. 

[Let  the  student  demonstrate  this  rule  by  remembering  that  y  =  cot  x  =  — — , 

Sill  >C 

and  also  by  taking  the  difference  between  the  consecutive  states  t/  =  cotx,  and 
y  -\-  dy  =  cot  (x  -}-  dx),  developing  and  reducing  as  in.  the  second  demonstration 
under  (69).] 

QUEKY. — What  is  the  significance  of  the  opposition  in  signs? 


The  differential  of  the  secant  of  an  arc  (or  angle)  is 
the  tangent  of  the  same  arc  into  its  secant  into  the  differential  of  the  arc. 

DEM.—  Let  M  —  sec  x  =s  •  —  -*.     Differentiating  by  (5&9  67)  we  have  dy  = 
cosx 


X  --  X  dx  =  tanxsecxdt,     Q, 

COb-  03  CObX          COS  05 


32  THE   DIFFERENTIAL   CALCULUS. 

72.  Prop. —  The  differential  of  the  cosecant  of  an  arc  (or  angle]  has 
the  opposite  algebraic  sign,  from  the  function,  and  is  numerically  equal  to 
the  cotangent  of  the  same  arc  into  its  cosecant  into  the  differential  of  the  arc. 

DEM. — Let  y  =  cosec x  =  sec  (90°  —  x).  Differentiating  by  the  last  proposition, 
dy  =  tan(90° —  x)sec(90° —  x)d(9Q° —  x)  =  cotxcosecx( — dx)  =  — cotxcosecxdx. 
Q.  E.  D. 

[Let  the  student  demonstrate  this  proposition  from  the  relation  y  =  cosec  x  = 

JL.i 

Bm;C 

QUERY. — What  is  the  significance  of  the  opposition  in  signs? 


73.  J*TOp. —  The  differential  of  the  versed-sine  of  an  arc  (or  angle} 
is  the  sine  of  the  same  arc  into  the  differential  of  the  arc. 

DEM. — Let  y  =  versx  =1  —  cosx.  Differentiating  by  (07),  dy  =  sinxdx. 
Q.  E.  D. 

QUERY.  — Why  should  the  differential  of  the  versed-sine  be  numerically  the  same 
as  the  differential  of  the  cosine,  but  have  an  opposite  sign  ?  Illustrate  geometri 
cally. 


74.  Prop. — The  differential  of  the  coversed-sine  of  an  arc  (or  angle) 
has  the  opposite  algebraic  sign  from  the  function,  and  is  numerically  equal 
to  the  cosine  of  the  same  arc  into  the  differential  of  the  arc. 

DEM. — (Similar  to  the  preceding.) 

QUERY. — Why  should  the  coversed-sine  have  the  same  differential  as  the  sine, 
but  with  an  opposite  algebraic  sign  ?  Illustrate  geometrically. 


EXERCISES. 

Ex.  1.  Differentiate  u  =  sin  x  cos  x. 

SUG. — Observe  that  we  have  here  the  product  of  two  variables,  viz.,  sinx  and 
cos  x.  Hence  da  =  cos  x  d(sin  x)  -f  sin  x  d(cos  x)  =  cos-  x  dx  —  sin2  x  dx  = 
(cos-x  —  sm-x]dxt  or  (2cos2x  —  l)dx,  or  (1  —  2sin2.r)dx,  or  cos2xdx. 

Ex.  2.   Differentiate  u  =  co$3x. 

SUG. — Observe  that  this  is  the  cube  of  the  variable  cos  a?.  Hence  apply  (56} 
and  we  have  du  =  3  cos2  x  d(cos  x)  =  —  3  cos2  x  sin  x  dx  =  3 (sin3  x  —  sin  x)dx. 

Ex.  3.  Differentiate  u  —  tan  5#. 
SUG.     du  =  sec2  5.-rd(5x)  =  5  sec2  5x  dx. 

Ex.  4.  Differentiate  u  =  cot2  x*.       du  —  — •  6.r2  cot  x3  cosec2  x3  dx. 

\ 

Ex.  5.  Differentiate  u  =  sin3  x  cos  x. 

du  —  sin2  x(3  —  k  sin3  ,r)«7x. 


TEIGONOMETRICAL  FUNCTIONS.  33 

Ex.  6.  Differentiate  u  =  3  sin<  x.  du  =  12  sin'  x  cos  x  dx. 

Ex.  7.   Differentiate  u  =  cos  mx.  du  =  —  m 

Ex.  8.  Differentiate  w  =  sin  3.r  cos  2#. 

du  =  (3  cos  3#  cos  2#  —  2  sin  3x  sin 

Ex.  9.  Differentiate  u  =  sec*5x.  du  =  10  sec*  5a?  tan  5xdx. 

Ex.  10.   Differentiate  u  =  tan"  nx.       du  =  w2  tan""1  w#  sec2  n#  cfo. 
Ex.  11.  Differentiate  u  =  log  sin  jr. 

SOLUTION.—  We  have  here  a  logarithm  to  differentiate,  i.  e.  the  logarithm  of  sin  a;. 
Hence  the  differential  is  the  differential  of  sin  a;,  divided  by  sin  a;,  in  the  Napier 

ian  system,  or  m  times  this,  in  the  common  system.     Therefore  du  —  md(sin  x^  — 

sin  x 
mcoKxdx 
—  -T—  -  —  =  m  cot  a;  dx,  or  in  the  Napierian  system,  cot  xdae. 

Ex.  12.  Differentiate  u  ==  log  cos  x. 

du  =  —  m  tan  x  dx,  or  —  tan  x  dx. 
Ex.  13.  Differentiate  u  —  lo    tan  x. 


du  = 


sin  x  cos  x 


sin  2 


tan  x 

Ex.  14.  Differentiate  u  =  log  cot  #. 
Ex.  1  5.  Differentiate  u  =  log  sec  x. 

Ex.  16.  Differentiate  M  =  log  cosec  x.  du  =  _  cot  x  dx. 

Ex.  17.  Differentiate  u  =  <7cos  x,  e  being  the  Napierian  base. 


SUG'S.      du    =    e*d(cosx)  -f-  cosxd(er)    =    --    ex  sin  a;  dx 
e*(cos  a?  —  sin  x}dx. 

Ex.  18.  Differentiate  it  =  xeco*x. 


SUG'S.      du    =     ecosrdx 
08Z(1  —  x  sin  a?)(fo. 


e*  cos  a  da- 


T»     irv    -r\-w        x-  x 

Ex.  19.  Differentiate  w  = 


—  COST 


=  e"  sin  ^  dx. 


Ex.  20.  Differentiate 


log  \/sin  a;  +  log  \/cos  or. 


SUG'S. 


i  log  sin  x  -f-  i  log  cos  x.     .  • .  du  =  J(cot  x  —  tan  x)dx  = 


dr 
tan  2s* 


THE  DIFFERENTIAL   CALCULUS. 


ILLUSTRATIVE    EXAMPLES. 

[NOTE. — The  object  of  these  examples  is  to  still  farther  illustrate  the  meaning  of  the  process  of 
differentiation.] 

Ex.  1.  Which  changes  the  faster  an  arc  or  its  sine  ?  "What  is  the 
relative  rate  of  change?  When  is  the  disparity  greatest  and  when 
least?  What  is  the  relative  rate  of  change  when  the  arc  is  60°  ? 
When  20°  ?  When  80°  ? 

SOLUTION. — From  y  =  sin  x,  we  have  by  differen 
tiating,  dy  =  cosxdc.  The  meaning  of  this  is,  that  if 
the  arc  (x,  AP)  takes  an  infinitesimal  increment 
(dp,  Pp)  the  sine  (y,  P/)  takes  an  infinitesimal  incre 
ment  (dy,  pE)  which  is  COSK  times  the  increment  of 
the  arc.  Now  cos  x  is,  in  general,  less  than  unity,  so 
that  the  increment  of  the  sine  is,  in  general,  less  than 
the  contemporaneous  increment  of  the  arc.  But  as  x 
grows  less  cos  x  becomes  greater  and  approaches  unity 
as  x  approaches  0.  So,  also,  cos.-r  approaches  0  as  x 
approaches  90°.  Hence  the  disparity  between  the  contemporaneous  increments 
of  an  arc  and  its  sine  is  less  as  the  arc  is  less,  disappears  when  the  arc  is  0, 
and  becomes  infinite  when  the  arc  is  90°.  For  x  =  0  we  may,  therefore,  con 
sider  the  arc  and  its  sine  to  be  increasing  at  equal  rates.  For  x  =  90°,  the  arc  is 
increasing  infinitely  faster  than  its  sine.  When  x  =  60°,  cos  x  =  i.  Hence  at  GO0 
the  sine  is  increasing  just  i  as  fast  as  the  arc.  In  the  figure,  letting  P'p'  represent 
dx,  />'E'  represents  dy  andp'E'  =  sP'p'.  When  x  =  20°  cos.r  =  .94  nearly. 
Hence  at  20°  the  sine  is  increasing  .9-1  as  fast  as  the  arc.  At  80°,  the  sine  is  in 
creasing  only  about .  17  as  fast  as  the  arc .  These  facts  are  illustrated  in  the  figure. 

Ex.  2.  Assuming  that  the  relative  rate  of  increase  remains  con 
stantly  the  same  as  at  40°,  how  much  does  the  sine  increase  when  the 
arc  increases  from  40°  to  40°  10'  ?  WThat  when  the  arc  increases  to 
41°? 


0  utf 


Suo.— Since  the  arc  of  10'  = 


3.14159 


.0029088  ;  we  find  the  increase  of  the 


180  X 

sine,  on  the  above  hypothesis,  to  be  .002228,  which  is  slightly  in  excess  of  the  real 
increase,  as  will  be  found  by  examining  a  table  of  natural  sines  in  which  the  de 
cimals  are  extended  to  7  places.  The  table  gives  .0022156. 

At  the  same  rate  of  increase  the  sine  of  41°  should  be  .013369  above  the  sine  of 
40°  ;  where;*;  from  a  table  the  increase  is  found  to  be  .0132714. 

[The  student  will  observe  that  the  cause  of  this  disagreement  is  that  the  rela 
tive  rate  of  increase  of  the  sine  as  compared  with  its  arc,  is  greater  at  40°  than  at 
any  point  between  40°  and  40°  10',  or  at  any  point  after  40°.] 

Ex.  3.  The  natural  tangent  of  27°  20'  is  .51G8755.  Assuming  that 
the  relative  rate  of  increase  of  the  tangent  as  compared  with  its  arc 


TRIGONOMETRICAL  FUNCTIONS.  35 

remains  the  same  as  at  this  point,  for  the  next  25"  increase  of  the  arc, 
what  is  the  natural  tangent  of  27°  20'  25"  ?  Am.,  .517029. 

Ex.  4.  "Which  increases  faster,  the  arc  or  its  tangent?  "When  is  this 
difference  greatest?  When  least?  What  is  the  value  of  the  arc 
when  the  tangent  is  increasing  just  twice  as  fast  as  the  arc? 

Answer  to  the  last,  45°. 

Ex.  5.  The  natural  cosine  of  5°  31'  is  .095308.  Assuming  that  the 
relative  rate  of  change  of  the  cosine  as  compared  with  the  arc  re 
mains  the  same  as  at  5°  31',  while  the  arc  increases  to  5°  32',  what  is 
the  cosine  of  5°  32'?  Ans.,  .995340. 

Ex.  6.  At  36°  what  is  the  relative  rate  of  increase  of  the  arc  and 
the  logarithm  of  its  tangent  ? 

SUG. — From  u  =  log  tan  x,  we  ha ve  du  =  m  (tana;  -f-  rotaOrie.  When  x  =  36°  this 
becomes  du  =  .43429  X  2.102925cfe  ;  or  the  logarithm  of  the  tangent  increases 
about  .91  times  as  fast  as  the  arc. 

Ex.  7.  The  logarithmic  cosine  of  67°  30'  is  9.582840.  Assuming 
that  the  relative  rate  of  change  of  the  logarithmic  cosine  and  the  arc 
remains  the  same  as  at  this  point  while  the  arc  passes  to  67°  31', 
what  is  the  logarithmic  cosine  of  the  latter  arc?  -4ns.,  9.582535. 

Ex.  8.  The  log  cot  58°21'  =  9.789868.  On  the  same  assumption 
as  above,  what  is  the  increase  of  this  logarithm  for  1  second  increase 
in  the  arc  ?  Ans.,  .00000471. 

Ex.  9.  The  log  cos  42°14'  =  9.869474.  What  is  the  corresponding 
tabular  difference  ? 

Ex.  10.  At  what  rate  relative  to  its  velocity,  is  a  point  in  the  cir 
cumference  of  a  wheel  revolving  in  a  vertical  plane,  ascending,  when 
it  is  60°  above  the  horizontal  plane  through  the  centre  of  motion? 

Ans.,  One  half  as  fast. 


CIRCULAR   FUNCTIONS. 

7£.  ~Prop.  —  The  differential  of  an  arc  in  terms  of  its  sine  is  the 
differential  of  the  sine  divided  Inj  the  square  root  of  1  minus  the  square 
of  the  sine  ;  or  the  differential  of  the  sine  divided  by  the  cosine. 

DEM.  —  Let  y  =  sin—  '#*,  whence  x  =  sin?/.     Differentiating  and  finding  the 


_  _ 

value  of  dy,  we  have  dy  =  —  —  .     But  cos  y  =  \/l  —  siu2*/  —  VI  —  x'2.    .  •  .  dy 


VI  — 


Q.  E.  D. 


*  This  notation  Is  explained  in  the  Trigonometry  of  this  series.     It  means  simply  "  y  =  tlie  arc 
whose  Bine  is  x,  and  hence  y  =  sm~'a;  is  equivalent  to  x  =  Bin  y." 


36  THE   DIFFERENTIAL   CALCULUS. 

70.  Sen. — The  student  should  not  fail  to  observe  the  essential  identity 
of  this  proposition  with  (06).  Thus,  when  we  differentiate  u  =  sin  a?,  we 
get  du  =  cos  xdx,  which  expresses  the  differential  of  the  sine  (u)  in  terms 

of  its  arc  (.r).     From  this  we  have  dx  =  =  — .  which  expresses 

cos.r       v/i_M* 

the  differential  of  the  arc  (x)  in  terms  of  the  sine  (u).     The  one  conception 
is  the  converse  of  the  other. 


77.  I*rop. — The  differential  of  an  arc  in  terms  of  its  cosine  has  the 
opposite  sign  from  the  function,  and  is  numerically  equal  to  the  differential 
of  fJi/}  cosine  divided  by  the  square  root  of  1  minus  the  square  of  the  co- 
sine ;  or  the.  differential  of  the,  cosine  divided  by  the  sine. 

DEM. — Let  y  =  cos— '  x,  whence  x  =  cos  y.  Differentiating,  and  finding  the  value 

dx  dx 

of  dy,  we  nave  dy  = : = —    — .     Q.  E.  D. 

sin  y  >/l  —  a* 

fS.  SCH  — Compare  this  and  the  following  propositions  with  their  equiv 
alents  in  Trigonometrical  functions,  as  was  done  in  the  case  of  the  preced 
ing  proposition. 


7.9.  J?vop. —  The  differential  of  an  arc  in  terms  of  its  tangent  is  the 
differential  of  the  tangent  divided  by  1  plus  the  square  of  the  tangent. 

DEM. — Let  y  =  tan— '  x,  whence  x  =  tuny.      Differentiating   and  finding  the 

value  of  dy,  we  have  dy  =  — - —  = — ,  since  sec2  y  =  1  4-  tan2  ?/  =  1  4-  x2. 

sec2  y        14-  ;e2 

Q.  E.  D. 


80.  J*rop. —  The  differential  of  an  arc  in  terms  of  its  cotangent  has 
the  opposite  sign  from  the  function,  and  is  numerically  equal  to  the  differ 
ential  of  the  cotangent  divided  by  1  plus  the  square  of  the  cotangent. 

DEM. — Let  y  ==•  cot—'  x,  whence  a*  =  coty.    Differentiating,  and  finding  the  value 

dx  dx 

of  dy,  we  have  dy  =  —  —   — -  =  —  - — • ,.     Q.  E.  D. 

cosec9  y  1  4-  x- 


81.  Prop. —  The  differential  of  an  arc  in  terms  of  its  secant  is  the 
differential  of  the  secant  divided  by  the  secant  into  the  square  root  of  the 
square  of  the  secant  minus  1. 

DEM.  — Let  y  =  sec-'  x,  whence  x  —  sec  y.     Differentiating  and  finding  the  value 

i>f  dy  we  have  dy  = — -,  since  tan  y  =  \/B60*  y  —  1  =  \/x'2  — 17 

~ 

Q.    E.   D. 


CIRCULAR   FUNCTIONS.  37 

82.  Prop. — The  differential  of  an  arc  in  terms  of  its  cosecant  has 
the  opposite  sign  from  the  function,  and  is  numerically  equal  to  the  differ 
ential  of  the  cosecant  divided  by  the  cosecant  into  the  square  root  of  the 
square  of  the  cosecant,  minus  1. 

DEM.— Let  y  =  cosec-1  x,  whence  x  =  cosecy.     Differentiating,  and  finding  the 

value  of  dif,  we  have  dy  = = .  since  coty  =  v/cosec*  u  —  1 

cosecycoty          «vV 1 

=  yV  —  1.       Q.  E.  D.  * 


S3.  Prof). — The  differential  of  an  arc  in  terms  of  its  versed-sine  is 
the  differential  of  the  versed-sine  divided  by  the  square  root  of  twice  the 
versed-sine  minus  the  square  of  the  versed-sine. 

DEM. — Let  y  =  vers~ lx,   whence  x  =  versy.     Differentiating  and  finding  the 


dx 


value  of  dy  we  have  dy  =  — — .     But  sin  y  =  v/1  —  cos2  y  =  \/L  —  (1  —  vers  y/-  = 
sin  y 


v/1  —  (1  — x)2  =  v%x  —  x2.     Therefore,  substituting,  dy  =  — — -  Q.  E.  r>. 

\/'lx  —  x* 

8£.  Prop. —  The  differential  of  an  arc  in  terms  of  its  coversed-sine 
has  the  opposite  sign  from  the  function,  and  is  numerically  equal  to  the 
differential  of  the  coversed-sine  divided  by  the  square  root  of  twice  the 
coversed-sine  minus  the  square  (f the  coversed-sine. 

DEM. — Let  y  =  covers~>  x,  whence  x  =  covers  y.    Differentiating,  and  finding  the 

dx  dx  dx 

Value  of  dy,  we  have  dy  =  —  = —  = — -  = 

COS2/  vxl  —  t>in5y  v/1  —  (1  — 

dx  dx 


~.       Q.  E.  D. 


EXERCISES. 

Ex.  1.  Differentiate  y  =  sin"1-  ;    y  =  cos"1-  ;    y  =  tan"1-  ;   y  = 

.r  x  _.x  _.x  ,x 

cot"1-  ;  y  =  sec"  -  ;  y  =  cosec    -  ;  y  =  vers    -  ;  y  =  covers    -. 


SUG.  —  We  have  dy  =  —  .-.  by  (7t>)-     Now  since  d[  -  )  =  —  ,  we  hare 


dx 


dy 


' 


=        __  =  —  --  .     In  like  manner  dy  =  d(  cos—1  '-  )  =  --  --  —  > 

I         X2        ^r~  —  x*  N  r/  Vr*  —  x* 


38  THE   DIFFERENTIAL   CALCULUS. 

rdx 


r*  J-  x* '     * 
1     i   * 

rdx  /  x\  rdx  /  x\  dx 

-  ;   dy  =  d(  cosec-i  -  )  =  — ;  dy  =  d(  vers-'  -  )  = ; 

JCY/  x~  —  r2  ^  *"'  x\/x*  —  r'z  ^  r'        \/2yx yi 


,  ,          _/  x\  dx 

and  dy  =  d[  covers—'  -  )  = 

V  r/  —    * 


dx 

—  x* 

GEOMETRICAL  ILLUSTRATION. — Let  OA  =  1,  and  OA' 
=  r.  Let  y  =  arc  C  A  (to  radius  1),  and  x  —  C'B'  the 
sine  of  the  same  number  of  degrees  as  y,  but  to  a  radius 

C'  B'        x 

r.     Now  CB  =  —     —  = '-,  and  we  have  y  (or  CA)  = 

sin—'  C  B    =   sin—'  —     —  =  sin—'  -,  the   arc    (y)   being 
taken  to  radius  1  while  the  sine  x  is  taken  to  the  radius  r. 

,x 


Ex.  2.  Differentiate       =  sin-1-  ;        ==  cos'1-  ;    ~  =  tan-1-  ;       = 

r  r  r  r      r  r      r 

,T     y  x     ?/  x  y  x  y  x 

cot    -  ;  -  =  sec    -  :  -  =  cosec"  -  ;  -  =  vers    -  ;  and  -  ==  covers"  -. 

r     r  r     r  r  r  r  r  r 

rdx  rdx 

liesults  in  order  :  dy  =  -  -      —  :  dy  =  --  —  —  ;  du 

~  ~ 


r"dx  r^dx  r"dx 

dy  =  --  •  ;    dy  =  -  —       -  ;  dy  =  ---  -         -  ;  dy  = 
' 


rdx  rdx 

;  and  dy  = 


\/'2rx  —  x* 

GEOMETRICAL    ILLUSTRATION. — In    F'KJ.    15   let    OA'   =   r,    C'A'   =  y,    and 

C'B'   =  x.      Now  if   OA  =  1,  we  have  CA  =  -,  and  CB  =  -.     Hence 

r  r 

dy  =  d(sin— *  x)to  radius  r  =  — — • .  etc. 


85.  Sen. — The  results  in  the  last  example  will  be  seen  to  correspond  with 
those  in  Ex.  1,  by  noticing  that  in  Ex.  1  y  represents  CA,  whereas  in 
Ex.  2  it  represents  C'A'.  Now  an  increment  of  C'B'  (which  is  x  in  both 
cases)  which  makes  an  increment  in  C  A,  will  make  r  times  as  great  an  incre 
ment  in  C'A'.  Hence  we  have  but  to  multiply  the  increments  of  CA  (the 
</y's)  as  found  in  Ex.  1,  by  r  to  get  the  corresponding  increments  in  C'A', 
which  are  the  cfy's  in  Ex.  2. 

/*•  7/^7  7*  '/Y/J,/ 

Ex.  3.  Differentiate  u  =  tan""1-.  du  =  '•?— • — . 

y  y*  -j-  x* 

Ex.  4.  Differentiate  u  =  sm~1(2jc\/l  —  a:8).  du  =  '  — . 

vl  —  #2 


CIRCULAR  FUNCTIONS.  39 

Ex.  5.  Differentiate  u  =  cos-\aVl  —  x*). 

SUG'S.—  By  the  rule  the  differential  of  the  arc  u  is  negative,  and  numerically 
equal  to  the  differential  of  its  cosine,  xVl—x*'  divided  by  the  square  root  of  1  minus 
the  square  of  its  cosine.  The  differential  of  x  \/l  —  x2  is  ctes/1  —  x'2 _ 

V(i  —  x-  -f-  x^i  —  x~) 
Ex.  6.  Differentiate  u  =  sin"1          -=^.  du  =  - — r—_-. 

Ex.  7.   Differentiate  u  =  sin"1  (3^7  —  4#3).  du  = 


vT-5 

Ex.  8.  Differentiate  u  =  vers"1?/  —  VZry  —  y\  understanding  that 
vers"1!/  is  taken  to  radius  r. 


Ex.  9.  Differentiate  w  =  tarr^N/l  +  ^2  —  a:). 


2(1  +  *> 

h     ,    ^ 

Ex.  10.  Differentiate  w  =  log  *  —       -  +  ^tan"1^. 

Al  1  —  x 

dx 
SUG.     u  =  i  log  (14-*)  —  i  l°g  (1  —  a?)  - 


Ex.  11.  Differentiate  ?/  =  sin"1  mo:. 

, 

Ex.  12.  Differentiate     =  etan    x. 


277 

Ex.  13.  Differentiate  ;*/  =  tan"1^ — ^: — -.  dy  = 


- 

J.    -(-   X* 


Ex.  14.  Differentiate  y  =  a 

i 

.  _i    ( .77lo^a7  +  (1  —  ^2)2  sin"1^ } 
dy  =  afln    *  -J -^—        -  j-  dx. 

™/1  ~.o\"2"  ^ 


GENERAL    SCHOLIUM, 

86.  The  preceding  sections  comprise  the  fundamental  rules  of  the  differ 
ential  calculus  ;  and  it  only  remains  to  extend  and  apply  them,  in  order  to 
complete  this  portion  of  our  subject. 


THE   DIFFERENTIAL    CALCULUS. 

SECTION  IV. 
Successive  Differentiation  and  Differential  Coefficients, 


SUCCESSIVE    DIFFERENTIATION. 

#7.  DEF. — Successive  Differentials  are  differentials  of  dif 
ferentials  ;  or  a  successive  differential  is  the  difference  between  two 
consecutive  states  of  a  differential. 

IL-L. — Let  M  N  Fig.  16,  be  a  straight  line  whose 
equation  is  y  —  ax  -f-  b  ;  whence  dy  =  adx.  Now 
suppose  x  to  be  considered  equicrescent,  and  let 
DD',  D'D",  D"D"',  and  D"'  D'v  represent  the 
successive  equal  increments.  P'E,  P"E',  P'"E", 
and  P'VE'"  represent  the  contemporaneous  incre 
ments  of  y,  i.  e.  the  dy's.  But  in  this  case  the  dy's 
are  all  equal.  Hence  there  being  no  difference  be 
tween  two  successive  states  of  dy,  as  between  P'E 
and  P"E',  there  is  no  successive  differential,  or  the 

differential  of  dy  is  0,  since  dy  is  constant.  This  fact  appears  also  from  the  rela 
tion  dy  =  adx,  in  which,  if  we  conceive  dx  to  be  constant  (i.  e,,  x  equicrescent), 
adx  is  constant ;  whence  dy,  which  equals  adx,  is  constant. 

But  consider  in  a  similar  manner  the  parabola  in  Fig.  17, 


FIG.  16. 


Still  con- 


whose  equation  is  y'2  —  2px  ;  whence  dy  =  - — . 

sidering  dx  as  constant,  i.  e.  DD'  =  D'D"  =  D"D'" 
=  D'"D1V,  etc.,  it  is  evident  that  the  dy's,  which  are 
represented  by  P  E,  P"E',  P'"E",  etc.,  are  not  constant. 
Now  the  difference  between  any  two  successive  values  of 
dy,  as  between  P'E  and  P"E',  is  a  successive  differ 
ential,  i.  e.  a  differential  of  a  differential.  The  fact  that 
dy  is  a  variable  in  this  case  when  dx  is  constant  is  also 


FIG.  17. 


readily  seen  from  its  value  dy  =^— .     In  this  pdx  is  constant,  but  y  is  variable. 
Hence  dy  varies  inversely  as  y. 

88.  DEF. — A  Second  Differential  is  a  differential  of  a  first 
differential,  is  represented  by  d2y,  and  read  " Second  differential  y" 
A  Third  Differential  is  a  differential  of  a  second  differential, 
is  represented  by  d*y,  and  read  "  Third  differential  y."  In  like  man 
ner  we  have  fourth,  fifth,  etc.,  differentials. 

Sen. — The  student  should  be  careful  not  to  confound  d-y  with  dyz.  The 
latter  is  the  square  of  dy.  Nor  should  the  superior  2  in  d-y  be  mistaken 
for  an  exponent :  it  has  no  analogy  to  an  exponent.  Observe  the  significa- 


SUCCESSIVE  DIFFERENTIATION.  41 

tion  of  the  several  expressions  c?9y,  dy*  and  d(yz}.     The  latter  is  equivalent 
to  tydy. 

SO.  JPTOp» — Second  differentials  are  formed  by  differentiating  first 
differentials,  third  differentials  by  differentiating  second  differentials,  etc., 
according  to  the  rules  already  given. 

This  proposition  is  self-evident,  since  the  differentials  are  expressed  as  algebraic, 
trigonometric,  logarithmic,  or  exponential  functions,  the  rules  for  differentiating 
which  are  those  heretofore  given. 

Ex.  1.  Produce  the  several  successive  differentials  of  y  =  ax*. 

SOLUTION.— Differentiating  y  =  ax*,  we  have  dy  =  4ax^dx.  Differentiating  this 
differential  remembering  that  d(dy),  i.  e.  the  differential  of  dy  is  written  d-y,  and 
that  dx  is  constant,  we  have  d-y  —  I2ax"-dx  dx,  or  12ax2  dxn~.  In  like  manner  dif 
ferentiating  d°y  =  12ax?dz2,  we  have  d*y  =  2±ax  dx3 .  And  again  d4y  =  24adx4. 
Here  the  operation  terminates,  since  d4y  being  equal  to  24adx4  is  constant. 

Ex.  2.  Produce  the  several  successive  differentials  of  y  =  Sx4  — 


«/ 

Ex.  3.  Produce  the  first  six  successive  differentials  of  y  =  sin  x. 

f  dy  =  cos  x  dx,  d*y  =  • —  sin  x  dx2, 
Results,   I  d3 y  =  —  cos  x  dx3,  d-y  =  sin  x  dx*, 
{  d^y  =  cos  x  dxr>,  dcy  =  —  sin  x  dx6. 

QUEBY. — Does  the  above  process  ever  terminate? 

Ex.  4.  What  is  the  3rd  differential  of  y  =  x*  ? 

I  . 

Ex.  5.  Produce  the  4th  differential  of  y  =  ax2. 

15a  dx* 


Ex.  6.  Produce  the  first  six  successive  differentials  of  y  =  cos  x. 
Ex.  7.  Produce  the  first  four  successive  differentials  of  y  =  logx, 
in  the  common  system. 

mdx                   mdx*    .          2mxdx*      2m  dx* 
Results,  dy  =  -  --,  d*y  = — ,  d*y  = -—=-—--,  d+y  = 


Gm  dx* 
x* 


42  THE   DIFFERENTIAL   CALCULUS. 

Ex.  8.    Produce   the   first    four    successive    differentials   of  y  — 
log  (1  +  a;),  in  the  common  system. 

mdx  m  dx"~  2ml+#    dx3       2m  dx3 

fiesults,  dy  ==  —  —  ,d2y  = 


~       + 
Ex.  9.  Produce  the  fourth  differential  of  y  =  er. 


Ex.  10.  Produce  the  fourth  differential  of  y  =  ax,  in  the  common 

log-4  a 

system.  d*y  =  —  £—  axda;4. 

J          m4 


DIFFERENTIAL    COEFFICIENTS. 

.90.  DEFS.— A  First  Differential  Coefficient  is  the  ratio  of 
the  differential  of  a  function  to  the  differential  of   its  variable,  and  is 

represented  thus,  — ^-,  y  being  a  function  of  the  variable  x. 

A   Second  Differential  Coefficient  is  the  ratio  of  the 
second  differential  of  a  function  to  the  square  of  .the  differential  of 

its  variable,  and  is  expressed  thus,  -~. 

A  Third  Differential  Coefficient  is  the  ratio  of  the  third 
differential  of  the  function  to  the  cube  of  the  differential  of  its  vari- 

cfoy 

able,  and  is  represented  thus,  -7-^.     In  like  manner  the  nth.  differen 
tial  coefficient  is  -~. 
dxn 

ILL.— Having  y  —  aas5,  we  obtain  -£  =  Sao;4.    In  strict  propriety  --  is  a  symbol 

representing  the  general  conception  of  the  ratio  of  an  infinitesimal  increment  of 
the  function  to  the  contemporaneous  infinitesimal  increment  of  its  variable  ;  and 
5ax4  is,  in  this  case,  its  value.  But  it  is  customary  to  speak  of  either  as  the  dif 
ferential  coefficient.  The  appropriateness  of  the  term  differential  coefficient  arises 
from  the  fact  that  the  5ax4  is  the  coefficient  by  which  the  differential  of  the  vari 
able  has  to  be  multiplied  in  order  to  produce  the  differential  of  the  function. 
Strictly,  therefore,  the  differential  coefficient  is  the  coefficient  of  the  differential 
of  the  variable  ;  but  it  is  customary  to  speak  of  it  as  the  differential  coefficient 
of*  the  function. 

*  The  "of"  meaning,  perhaps,  "derived  from,"  or  "appertaining  to." 


DIFFERENTIAL   COEFFICIENTS.  43 

Ex.  1.  Given  y  =  ax3  —  xz,  to  find  the  1st,  2nd,  and  3rd  differential 
coefficients. 


Hesults,         =  3ax*  —  2ar,         =  Gax  —  2,         =  Go. 
dx  dx*  dx* 

1  4-  x 
Ex.  2.  Given  y  =  —     --,  to  find  the  5th  differential  coefficient. 

d*  240 


dx*        (1  —  x 
Ex.  3.  Given  y  =  e?*,  to  find  the  4th  differential  coefficient. 


Ex.  4.  Given  y  =  ax,  to  find  the  5th  differential  coefficient. 

~-  =1=  log5  a  ax. 
dx'a 

Ex.  5.  Produce  the  first  three  successive  differential  coefficients  of 
y  =  tan  x. 

Results,  ~  =  sec2  07.     — ^  =  2  sec2  #  tam*.    -—    =  4  sec2  a?  tan2  x  4- 
dx  dx*  dx'A 

2  sec4  x. 

[NOTE. — The  10  examples  in  the  former  part  of  this  section  may  be  used  for  further  illustration 
of  this  subject,  if  desired.] 

Ex.  G.  Given  y2  =  2pa?,  to  form  the  third  differential  coefficient. 

SOLUTION. — Differentiating  and  finding  the  coefficient,  we  have  --  =  -.     To  dif- 

clx       y 

ferentiate  ---  we  have  but  to  remember  that  dy  is  a  variable,  that  its  differential  is 
dx, 

d-ii,  and  that  dx  is  constant  and  hence  remains  the  same  ;  whence  d(  ---  )  =  —^-. 

\dx/         dx 

Differentiating  -,  we  have  d(~  )  =  —  —  -.     Hence  —^  =  —  -— .     But  the  second 

y  \2//          y'  dx          y* 

differential  coefficient  is  the  ratio  of  the  second  differential  of  the  function  to  the 
square  of  the  differential  of  its  variable.     Hence  dividing  by  dx,  we  have   — ^  = 

Pl  P~ 

—  ~— .     But  --  =  -.     Substituting  this  value,  -^-  =  —  —  =  —  — .     In  a  similar 
y2  dx       y  dx2  y2  yA 

_  d?-y      3p3 

manner  we  find  -—  =  -£-. 
dx*        y* 

d*y  d*y  S4 

Ex.  7.  From  A*y*  +  B-xz  =  AZB2,  find  -^-.  -^-  — 

d&  dx* 

d2y       %c9 

Ex.  8.  From  xy  =  c2  show  that  --  =  — . 

dx2        x3 


44  THE   DIFFERENTIAL   CALCULUS. 

Ex.  9.  From  y  =  —?—-  show  that  f y-  = 

*?  .-v»  >-7  /v>  J 


(1  —  a;)5 
Ex.  10.  From  yz  —  sec  2x  show  that  y  -f  ™  =  3*/5. 

(Z^l? 

[NOTE.  —  The  first  differential  coefficient  expresses  the  relative  rate  of  increase  of  the  function 
and  its  variable,  and  its  significance  is  illustrated  in  the  examples  on  pages  31,  35,  which  it  will  be 
well  to  review  for  this  purpose.] 

Ex.  11.  What  is  the  third  differential  of  u  =  ~,   dx  being  con- 

cix 

d4ii 

stant?  d3u  =  —  -. 

dx 

Ex.  12.  What  is  the  differential  of   --,  dx  not  being  considered 

das 

d"ydx  — 
constant?  Ans., 


dx* 

SCH.  —  Differential  coefficients  are  generally  variables  and  hence  can  be 
differentiated.  In  differentiating  them  it  is  important  to  observe  whether 
die  is  conceived  as  constant  or  variable.  To  say  that  the  first  differential 
coefficient  is  generally  variable  is  equivalent  to  saying,  in  the  case  of  a  curve, 
that  the  relative  rate  of  change  of  the  ordinate  and  abscissa  varies  for  dif 
ferent  points,  as  we  have  seen  heretofore. 


SECTION  V. 

Junctions  of  Several  Variables,  Partial  Differentiation,  and 
Differentiation  of  Implicit  and  of  Compound  Functions, 

01.  When  a  quantity  is  a  function  of  two  or  more  variables,  as 
u  =y(j7,  y),  these  variables  may  be  independent  of  each  other,  or 
dependent  upon  each  other.  This  distinction  is  marked  by  saying 
that  u  is  a  function  of  two  or  more  independent,*  or  two  or  more 
dependent*  variables  as  the  case  may  be. 

ILL.— Let  x  and  y  be  the  sides  of  a  rectangle  and  u  its  area.  Then  u  =  xy,  in 
which  w  is  a  function  of  the  two  independent  variables  x  and  y.  That  is,  x  may 
vary  without  causing  any  variation  in  y,  and  y  may  vary  without  causing  a  varia 
tion  in  x. 

But  suppose  u  to  represent  a  rectangle  which  is  to  remain  similar  to  a  given 
rectangle.  Now  when  x  or  y  changes,  the  other  must  change  also,  and  we  have  u 
a  function  of  two  dependent  variables. 

*  These  terms  are  here  used  in  a  slightly  different  sense  from  that  hitherto  assigned  them. 


PARTIAL  DIFFERENTIATION.  45 

92.  DEF.— A  Partial  Differential  of  a  function  of  two  or 
more  variables  is  a  differential  taken  under  the  hypothesis  that  one 
or  more  of  the  variables  remains  constant ;  usually  we  consider  the 
partial  differential  under  the  hypothesis  that  but  one  of  the  variables 
changes,  the  others  remaining  constant. 

93.  DEF.— A   Total  Differential  of  a  function  of  two  or 
more  variables  is  a  differential  taken  under  the  hypothesis  that  all  the 
variables  upon  which  it  depends  vary. 

ILL.— If  u  =  3ax??/  —  2y2  -f-  3bx*  —  5,  and  x  and  y  are  conceived  as  entirely 
independent  of  each  other,*  it  is  evidently  legitimate  to  inquire  what  effect  upon  u 
will  be  produced  by  a  change  of  x  or  y  alone.  Thus,  if  we  suppose  x  to  take  an 
infinitesimal  increment  and  y  to  remain  unchanged,  the  contemporaneous  incre 
ment  of  u  is  du  =  (Gaxy  -f  96^)dx.  If,  on  the  other  hand,  y  takes  an  infinitesi 
mal  increment  and  x  remains  constant,  the  contemporaneous  increment  of  u  is 
du  =  (3«x2  —  ±y)dy. 

But  how  is  it  when  x  and  y  are  dependent  upon  each  other?  Evidently,  if  we 
wish  to  obtain  the  total  change  in  u  due  to  a  change  in  both  x  and  y,  we  may  still 
conceive  them  to  change  in  succession.  Only,  in  this  case,  we  must  remember  that 
the  change  in  y  (for  example)  is  not  independent  of  the  change  in  x.  That  is, 
that  dy  is  a  function  of  dx,  which  is  not  the  case  under  the  former  hypothesis. 

94.  DEF.— A    Partial   Differential    Coefficient  is  the 

ratio  of   a  partial  differential  of   a  function  of  several  variables,  to 
the  differential  of  the  quantity  supposed  to  vary. 

93.  DEF.—  A  Total  Differential  Coefficient  of  a  func 
tion  of  two  or  more  variables  is  the  ratio  of  the  total  differential  of 
the  function  to  the  differential  of  some  one  of  the  variables  ;  and 
there  may  be  as  many  such  coefficients  as  there  are  variables. 

96.  SCH. — When  several  independent  variables  enter  into  a  function,  we 
might,  if  we  chose,  consider  each  of  them  equicrescent,  and  its  differential 
constant ;  but  when  the  variables  are  mutually  interdependent,  it  is  evident 
that  in  general  but  one  can  be  considered  equicrescent. 

#7.  Prop.— The  total  differential  of  a  function  of  several  variables 
is  equal  to  the  sum  of  the  partial  differentials. 

DEM.—  Let  u  =f(x,  2/).  Represent  the  partial  differential  of  u  with  respect  to 
x,  by  dxu  and  with  respect  to  y  by  dyu,  while  du  represents  the  total  differential. 
Now  du=^f(x  +  dx,  y  +  dy)  — /(#,  y)  ;  I  e.  it  is  the  difference  between  two  con 
secutive  states  of  the  function  when  both  x  and  y  have  taken  increments.  Now 
subtract  and  add  f(x  +  dx,  y),  which  will  not  change  the  value,  and  we  have 

du  =f(x  -f-  dx,  y  +  dy)  — /(x  -f  dx,  y)  -f/(x  -f  dx,  y)  — /(x,  y). 
But  f(x  -f  dx,  y  -\-  dy)\  —~f(x  +  dx,  y)+  =  dvu,  since  it  is  the  difference  between 

"  *  Something  in  the  nature  of  the  problem  always  determines  whether  x  and  y  are  dependent  or 
independent. 

t  The  student  will  notice  that,  however  x  and  y  are  involved,  the  only  difference  between 
J\x  -f  dx,  y  +  dy)  and  J\x  +  dx,  y)  is  what  arises  from  y  passing  to  y  -f  dy. 


46  THE  DIFFERENTIAL   CALCULUS. 

two  consecutive  states  of  the  function  due  to  a  change  in  y  alone.  So,  also, 
f(x  -f-  dx,  y}  —  f(X,  y)  =  dxii,  for  a  like  reason.  .  • .  du  =  dj-u  -\-  duu  ;  and  it  is 
evident  that  a  similar  course  of  reasoning  can  be  applied  when  u  is  a  function  of 
any  number  of  variables.  Q.  E.  D. 

ILL. — This  proposition  may  be  perplexing  to  a  thoughtful  student,  even  after 
he  has  learned  the  demonstration.  The  following  illustration,  for  the  substance 
of  which  I  am  indebted  to  Price,  may  help  him  to  realize  its  truth.  Consider  the 
amount  of  grain  grown  upon  a  piece  of  land.  This  is  evidently  a  function'  of  the 
area,  the  soil  (quality  of),  and  the  Ullage,  to  say  nothing  more.  Let  G  represent 
the  total  amount  of  grain,  A  the  area,  8  the  soil  (quality  of),  and  T  the  tillage. 
In  mathematical  notation  we  have  G  =f^A,  8,  T).  Now  consider  A,  8,  and  Tas 
variable. 

1st.  We  may  inquire  what  effect  upon  G  will  be  produced  by  an  increment  (for 
our  present  purpose  infinitesimal)  of  A,  while  8  and  T  remain  constant.  This 
will  give  a  partial  differential  of  G  with  respect  to  A.  In  like  manner  we  may 
inquire  what  effect  upon  G,  an  infinitesimal  increment  of  8,  or  infinitesimal  incre 
ments  of  A  and  8,  will  produce.  And  again  what  effect  will  be  produced 
by  a  change  in  any  one,  any  two,  or  in  all  three  of  the  variables,  A,  8,  and  T. 
The  latter  would  be  the  total  differential  of  G. 

2nd.  It  is  evident  that  A  is  independent  of  S,  and  T,  while  8  and  T  are  (proba 
bly)  dependent  upon  each  other,  i.  e.  good  tillage  may  have  a  larger  proportionate 
effect  upon  the  crop  on  good  soil,  than  upon  poor. 

3rd.  If  we  consider  the  effect  upon  G,  produced  by  an  infinitesimal  increment 
of  A,  while  £and  Tare  constant,  calling  this  dAG  ;  and  then  (not  considering  A 
as  having  increased)  consider  the  effect  on  G  of  an  infinitesimal  increment  of  8, 
calling  it  dsG  ;  and  then  (not  considering  either  A  or  8  as  having  increased)  con 
sider  the  effect  upon  G  produced  by  an  infinitesimal  increment  of  T,  calling  it 
drG  ;  it  may  seem  that 

dG  =  dAG  4-  dsQ-  +  dTG 

wall  not  represent  the  total  change  in  G  when  A,  8,  and  T  change  together.  For 
example,  we  lose  the  effect  of  the  improvement  in  soil  and  tillage  upon  the  incre 
ment  of  the  area,  and  also  the  effect  of  the  increased  effect  of  better  tillage  upon 
the  improvement  of  the  soil.  But  it  is  easy  to  see  that  these  effects  are  infinites 
imals  of  infinitesimals,  and  that  the  effects  we  seek  to  trace  are  infinitesimals  of 
the  first  order,  hence  the  former  must  be  omitted. 

ANOTHER  ILLUSTRATION  to  the  same  purpose  is  furnished 
by  the  parallelopipedon.  Let  u  represent  the  volume  of 
the  parallelopipedon  indicated  by  the  dotted  lines,  of 
which  x,  y,  and  z  are  the  length,  breadth,  and  height, 
respectively.  Then  u  =  f(x,  y,  z)  =  xyz.  The  additions 
represented  upon  the  end,  side,  and  top  respectively,  are  -p 

eZ,u,  dyU,  and  dzu,  but  it  does  not  appear  that  the  sum  of 

these  make  up  the  total  increment  of  u  due  to  an  increase  of  all  three  of  the  vari 
ables.  But  it  cZoe.s  appear  that  the  wanting  parts  will  be  infinitesimals  of  the 
second  and  third  orders,  when  the  increments  of  x,  y,  and  z  are  infinitesimal,  and 
as  the  increments  represented  in  the  figure  are  infinitesimals  of  the  first  order, 
the  others  must  be  omitted,  in  relation  to  the  latter. 

Ex.  1.  On  the  principle  of  the  above  proposition  differentiate  u  = 


PARTIAL   DIFFERENTIATION.  47 

3ar:t/2  —  5x*  —  2y  — 10.     Observe  also  that  the  result  agrees  with  that 
obtained  by  the  method  before  learned. 

SOLUTION.— Differentiating  with  respect  to  x,  we  have  d,u  =  Gxy*dx — lox^dx. 
Again,  differentiating  with  respect  to  y,  we  have  dyu  =  6x~ydy  —  2dy.  Adding, 
du  =  dxii  4-  dvu  —  Gxy-dx  —  15a?dx  4-  Gx-ydy  —  2dy.  Finally,  differentiating  by 
the  elementary  methods,  du  =  Gxy'dx  4-  Gx'ydy  —  15xydx  —  My,  a  result  identi 
cal  with  the  preceding. 

Ex.  2.  Differentiate  u  =  xv  both  by  the  above  principle  and  by 
passing  to  logarithms  and  using  the  elementary  methods,  and  com 
pare  the  results. 


SUG'S.     djU  =  yx'J~ldx.      duii  =  xj>  log  x  dy.      .'.    du  =  yxy—ldx  4-  xv\ogxdy. 

du          dx  ,  uydx    . 

log  u  =  ylogx.     Hence  —  =  y  --  \-logxdy,  ordu  =  --  --  \-  ulogxdy  = 

11  X>  X 

' 

-j-  xy  log  x  dy  =  x"-ly  dx  -J-  x'J  log  x  dy. 

Ex.  3.  Differentiate  u  =  tan"1-,  both  by  the  method  of  partial  differ 
entials  and  by  the  elementary  method. 

ydx  dy 

/>»2  7/r/T*  X* 

SUG'S.  —  By  partial  differentiation,  dxu  —         "  —  =  --   ,'    -.     d,,u  =      '        = 

i  +  S      +ffl       !+"; 

xdy  —  ydx 


<y\ 
x) 
.  •  .  au  =  —  -  -  —  .  ±>y  tne  elementary  metnoa,  au  =  --  -  = 


_  xdy  —  ydx 
~x-2  4.  yF"  • 

[NOTE. — The  pupil  will  doubtless  be  led  to  inquire,  Why  use  the  method  of  partial  differentiation 
when  the  elementary  methods  seem  to  be  more  expeditious?  It  is  not  for  its  use  in  such  ele 
mentary  processes  that  it  is  valuable.  These  examples  are  given  only  to  illustrate  the  propo 
sition  ;  the  utility  of  it  will  appear  hereafter.] 

Ex.  4.  Differentiate  as  above  u  = — . 


SUG'S.     djcU  = '- = • •     dyU  = '— 

(x  —  yf  (x  —  y)*  ( 

%(xdy  —  ydx} 
-.     .' .  du= : : . 


Ex.  5.  Differentiate  as  above  u  —  sin  (xy). 

du  —  cos  (xy)  [ydx  4-  xdy]. 
Ex.  6.  Differentiate  as  above  u  =  log  x\ 

du  =  d.fu  4-  dvu  =  '— -  4-  log  x  dy. 


48  THE  DIFFERENTIAL  CALCULUS. 

Ex.  7.  Differentiate  as  above  u  =  y*in*. 

du    =    dxu    -f-    dvu    =    yam  x  log  y  cos  x  dx   -f-   sin  x  y81"  *~ldy 

y*m  x  jOg  y  cog  x  dx  +  sin  a;  cove,.s<« 
Ex.  8  .  Differentiate  as  above  u  =  vers"1  -. 

y 

x  dy 
_          dx  y  ydx  —  xdy 


Ex.  9.  Differentiate  as  above  u  =  sin  (.x  -f-  y}- 

du  =  cos  (x  -f 
Ex.  10.  Differentiate  as  above  u  — 


.9$.  dotation. — Since  when  u=f(x),  du=^=  the  first  differential 
coefficient  of  u  with  respect  to  x,  multiplied  by  dx,  we  may  symbolize 

dxu,  by  — -dx.     So  also  dyu  =  -j-dy. 
J  dx  dy 

00.  I*rop. —  The  formula  for  the  total  differential  coefficient  of  u 
with  respect  to  x,  when  u  =  f(x,  y)  is 

rdu-\          du        du  dy 
\_dx\          dx        dy  dx' 

in  which  —,  and  -—  are  partial  differential  coefficients,  and  the  [  ]  indi 
cate  the  total  coefficient. 

DEM. — We  have  du  =  dxu  4-  dvii  =  —  dx  4-  —  dy.     Dividing  by  dx,  and  dis- 

dx        •    dy 

tinguishing  the  total  differential  coefficient  by  the  [  ],  we  have  I  —  I  = 1 -. 

[_dx  J       dx  '  dy  dx 
Q.  E.  D. 

SCH. — This  formula  has  definite  meaning  only  when  y  and  x  are  mutually 
dependent,  i.  e.  when  y  =f(x),  since  otherwise  —  is  indeterminate.  The 

formula,  therefore,  signifies  that  when  u  =f(x,  y},  and  y  —  /i  (x),  the  total 
differential  coefficient  of  u  with  respect  to  x  is  equal  to  the  partial  differ 
ential  coefficient  of  u  with  respect  to  x,  -f-  the  product  of  the  partial  differ 
ential  coefficient  of  u  with  respect  to  y,  multiplied  by  the  differential  co 
efficient  of  y  with  respect  to  x,  obtained  from  the  relation  y  =fl  (x). 

The  following  language  is  often  used  to  express  the  relation  of  u  to  x  and 
y  in  such  a  case  ;  viz. ,  u  is  directly  a  function  of  x,  and  indirectly  a  function 
of  x  through  y. 

rdu-\        du        du  dy 

It  might  seem  that  the  relation  = 1 -is  absurd,  since  by 

LdxJ        dx        dy  dx 


FUNCTIONS   OF   SEVERAL  VARIABLES.  49 

r~du~\          du 

cancelling  the  dy,  it  reduces  to        -    =  2  —  .     But  this  is  to  misapprehend 

LdxJ         dx 

entirely  the  significance  of  the  notation.     It  is  to  be  observed  that  the  du 

du   .    ,  ,     du         ,     r~du~\ 

in  --,—  ,  is  by  no  means  necessarily  the  same  as  the  du  in  —  ,  or  in  I  —   .     In 
dx  dy  \_dx\ 

--,  du  is  the  increment  of  u  due  to  an  infinitesimal  increment  of  x  in  the 
ax 

function  u  =  f(x,  y),  while  y  remains  constant.     In  like  manner  the  du  in 

—  .  is  the  increment  of  u  due  to  an  infinitesimal  increment  of  y,  while  x 
dy 

remains  constant.  These  will  by  no  means  be  generally  the  same.  Much 
less  will  either  of  these  du's  be  the  same  as  the  du  in  [%-~L  which  is  the 
change  in  u  due  to  a  change  in  both  the  variables  x  and  y.  Finally,  there 
is  nothing  in  the  logic  of  the  process  by  which  the  expression  —  -  —  was 

arrived  at,  that  makes  the  dy's  in  it  equal  to  each  other.  The  dy  in  —  is 
simply  an  arbitrary  infinitesimal  increment  assigned  to  y  in  u  =  f(x,  y},  x 
remaining  constant  ;  while  dy  in  -~  is  the  increment  which  y  takes  on  in 
the  function  y  =  fl  (x},  when  x  takes  the  increment  dx. 

rdu-i        du       du  dy 
100.  Con.  1.  —  From  u  =.  f  (x,  y,  z)  we  have  I  —    =  —  +  —  -^-  -f 

du  dz    . 

—  -—  ,  in  which  y  and  z  are  junctions  of  x. 


DEM.  —  Since  the  total  differential  equals  the  sum  of  the  partials,  we  have 

du  ,      ,    du  .      ,    du  ,  rdu~\        du    ,    du  dy    ,    du  dz 

du  =  --dx  -f  —dy  -f  --dz.      Dividing  by  dx,  \  =  -,-  +  -,-  -f  -f  -r-  -=-. 

dx        '    dy  y    '    dz  LdxJ        dx    '    dy  dx    r    dz  dx 

Q.  E.  D. 


Sen.  —  In  the  same  manner  the  total  differential  of  a  function  of  any 
number  of  variables  dependent  or  independent,  may  be  found  ;  and,  when 
all  are  dependent  upon  some  single  variable,  the  differential  coefficient 
with  respect  to  that  one  may  be  formed. 

101.  COE.  2.  —  If  u  =  f(y,  z,  w),  and  y  =  <p(x),  z  =  ^(tf),  and 
du  _  du  dy       du  dz       du  dw 


[Let  the  student  give  the  proof.] 

Ex.  1.  Given  u  =  tan"1-,  and  y"*  -f-  &  =  ^2>  to  find  I  --  I. 

•7 

rdu~]        du    .    du  dy  .  du         du 

SOLUTION.  —  We  have     —     =•  ---  ----  '-.     Eemembenng  that  —  and  --  are 

LdxJ        dx  ~  dy  dx  dx         dy 


50  THE  DIFFERENTIAL  CALCULUS. 

/-)  — 

partial  differential  coefficients,  we  have  from  u  =  tan—1-,  d  u  =  ••»« 

'    * 


=  —  —  ;  whence  —  -=  —  .  In  like  manner  we  find  -r—  =  --  .  Also  from  w24-x2  =  r2, 
r*  cte     r2  dy          r-2 

find  (Jy  =  —  -.     Substituting  these  values,  we  obtain  f—  1  =  ^  4-  (  —  -  V—  -'\ 
dx  y  \_dxj       r*  ^\      r*A      y) 

_  y         x2  _  ?/2  -f  ,r2  _  r2  _  1  1 

—   ,.3  r.y   —  ,-iy  —  ^y   —   y'    ^    ^   __  xt 

Ex.  2.  Given  u  =  tan"1^)*  and  y  =  f?  to  form  f-r-|. 

LwiT/'  J 

t    rdu-\        e*(l  +  x) 
Result,     —    =  ---  --  £. 
LdxA        1  +  #2e2a: 

Ex.  3.  Given  w  =  z2  +  2/3  +  zy,  and  z  =  sinor,  y  —  e*,  to  form  [-T-  !• 


SUG'S.—  We  have  =  --    --  4-  —  — 

dy  dx  ~  dz  dx 


+  t/)(cosx) 
(2sinic  -f 

=  3e3j:  -|-  ^(sino;  -j-  cosic)  -f-  2  sin  x  cos  x 
=  Se3-1  -f  e*(sin  x  +  cos  x)  -J-  sin  2x. 


Ex.  4.  Given  n  =  yz,  and  y  =  e%  z  =  #4  —  4a;3  +  12.r2  —  24a:  +  24, 
to  find  g]  .  «[£]  =  -- 

Ex.  5.  Given  u  =  sm~^(p  —  5),  andp=  3x,  q=4x*,  to  find  [  —  1. 

du  1  du  —I  dp 

SUGS.—  We  have  —  =  .        -    =    —  -  =z,     -~   =  3, 

dp         v/x  __       _     *       dq  ^/i  —       —      *      » 


and          =    12*.        Whence  =    -  __  __    -    — 

Ldx_\  ^l  „  (p  _  q)-z  v/i  —  (p  — 


Ex.  6.  Given  u  =  ^-  —  —  -f  ^-  and  i/  =  logx,  to  find  that  |—  1 
4  8        32  La^CJ 


Ex.  7.  Given  u  =  -  —  ^  ~-i,  where  p  =  a  sin  ar,  and  q  =  cos  #,  to 


-du 


find  that    — -  =  eaz  sin  x. 
Ldxj 


[NOTE.— In  such  examples  as  the  above,  it  is  of  course  possible  to  subst'tute  in  the  function  «, 
the  values  of  the  several  variables  on  which  it  depends,  in  terms  of  the  single  variable  upon 


FUNCTIONS   OF  SEVERAL  VARIABLES.  51 

which  each  of  them  depends,  and  then  have  u  =  a  function  of  a  single  variable,  which  can  be 
differentiated  by  the  elementary  processes.  But  it  is  the  chief  design  of  these  examples  to  fa 
miliarize  the  important  formula*,  used,  and  render  their  meaning  clear.  Their  precise  practical 
value  cannot  be  appreciated  until  the  student  has  made  further  progress  ] 


IMPLICIT   FUNCTIONS. 

du 

102.  Prop.— Having  f(x,  y)  =  0  =  u,  ^  =  —  ~  ;    in  which 

_    and  —  are  the  partial  differential  coefficients  of  the  function  taken 

dx'          dy 

with  reference  to  x  and  y  respectively. 

pEM. From  (99)  we  have  I  —  I  =  -= — |-  -r-  ~~'     ^u^  as  u  remams  constantly 

equal  to  0,  for  all  simultaneous  values  of  x  and  y,  when  both  x  and  y  have  taken 
on  contemp9raneous  changes,  du  =  0.    Therefore  jj^J  =  0,  and  ^  +  ^  ^|  =  0  ; 

du 

dv  dx 

whence  -f  =  —  —.     Q.  E.  D. 
dx  du 


SCH.  _  it  is  to  be  observed  that  though  |^J  =  0,  it  by  no  means  follows 

that  --,   or  ~  =  0.      For  example,  y*  +  aft  —  r*  =  0  is  of    the  form 
dx         dy 


__  Q  NOW  if  y  changes  and  x  does  not,  the  function  changes  and  is 
no  longer  =  0.  So  also  if  x  changes  and  y  does  not,  the  function  is  not  0. 
But  if  both  change  together  according  to  the  law  of  their  mutual  depend 
ence,  i.  e.  if  the  changes  are  what  we  have  called  contemporaneous,  the 
function  remains  equal  to  0  ;  and  its  total  differential  is  0.  [The  student 
can  observe  the  geometrical  signification  of  these  statements,  by  noticing 
that  yn~  -f  x*  —  r2  =  0  is  the  equation  of  a  circle,  and  that  the  function 
is  0  when  x  and  y  vary  together,  according  to  their  mutual  dependence  : 
but  when  one  varies  and  the  other  does  not,  the  function  varies  ;  i.  e.  the 
point  falls  out  of  the  circumference.  Illustrate  in  like  manner  from  y2  - 
2pa?  =  0.] 

Ex.  1.  Given  x*  +  2a#2?/  —  ay*  =  0,  to  form  ^|  upon  the  principle 

• 
just  demonstrated. 

SOLUTION.  -Putting  u  =  0  =  x*  +  2axny  —  ay\  we  have  dru  =  4&dx  +  4rt.ryd.r, 
and  dyU  =  2ax*dy  _  Zay^dy  ;   whence  we  have  the  partial  differential  coefficients 

du 
dy  dx  4,&  -f- 


52  THE  DIFFERENTIAL  CALCULUS. 

Ex.  2.  Given  axs  -f  x3y  —  ay3  =  0,  to  form  -~  as  above. 

dx 


i    dy  Saw?*  f  3j?2 

Result,  ~  = 


Ex.  3.  Given  if-  —  2axy  +  #2  —  52  =  Q,  to  form  -^  as  above. 

ox 


3ay< 

re. 
dy       a?/  —  .r 


J  dx        y  —  ax 


Ex.  4.  Given  ys  —  3y  +  x  =  0,  to  form  --  as  above. 


,  - . 

'  dx        3(1  —  y*) 


Ex.  5.  Given  xv  —  y*  =  0,  to  form  -^-  as  above. 

dx 


Result,  dJl  =  y- 


. 

a:2  —  ^-?/  log  x 

Ex.  6.  Given  a*"  +  v/sec(^r?/)  =  0,  to  form  --^  as  above. 


y1    sec  (art/)  tan  (.r?/) 


COMPOUND   FUNCTIONS. 

105.  DEF.  —  ^1  Compound  Function  is  a  function  of  a 
function.  Thus,  if  u  =  /(y),  and  y  =  ^(a?),  u  =  /[^(ar)],*  and  w 
is  said  to  be  a  compound  function  of  x.  This  relation  is  often  indi 
cated  by  saying  that  "u  is  a  function  of  x  through  y"  or  that  "  u 
is  indirectly  a  function  of  57  through  y." 

In  case  u  =  f(x,  y),  and  y  =  9(0?),  we  say  that  w  is  directly  a 
function  of  ar,  and  also  indirectly  through  y. 

104.  Prop.—  If  u  =  f(y)  and  y  =*  9,^),  ^  =  ^  g. 

DEM.—  From  u  ==  f(y\  we  have^w  =  -r1^.     But  from  y  =  tp(x)t  we  have  dy  = 

%  ,       TT  ,         ^'<  r?v  7          ,  dn       du  dy 

-f-dx.     Hence  du  =  —  -^-dx,  and  —  =  --  -.     o.  E.  D. 

rf*  dy  dx  dx      dy  dx 


It  will  be  seen  that  this  is  only  a  particular  case  of  the  preceding  ; 
but  it  is  of  such  frequent  occurrence  that  it  is  thought  best  to  give  it  prom 
inence. 

*  Head,  "u  equals  the/  function  of  the  tp  fimction  of  x." 


SUCCESSIVE  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  VARIABLES.      53 

Ex.  1.   Given  u  =  a\  and  y  =  b',to  form  -^  on  the  principle  just 
demonstrated. 


So™,     g  =  «,log«,   and  f  =  *,og&.     .,  |  =  ||  =.a,logo  x 

6*  log  I  =  a^b*  log  a  log  6,  or  afcV  log  a  log  6. 

Ex.  2.  Given  w  =  6y",  an(j  ^  =  a#s,to  form  —  as  above. 


cLx 

du 
— 
dx 


Ex.  3.  Given  u  =  logy,  and  ?/  ==  log#  to  form  -^  as  above. 

wJt7 

<7w        1     1  1 


.  Prop.  —  Having  given  u=  <p(z)  andz  =/(#,  y)  to  differen 
tiate  u  with  respect  to  x  and  y  without  previously  eliminating  z. 

DEM.  —  Since  u  is  a  function  of  a;  and  y,  we  have 


Now  w  is  a  function  of  x  through  z  (i.  e.  it  is  a  compound  function  of  x).     Hence 

du        du  dz  du       du  dz 

•T-  =  y-  7-;  and  fora  like  reason  —  =—    -  (1O4).     Therefore,   substituting, 

aa;        dz  dx  dy      dz  dy  ^ 

,  du  dz  du  dz  , 

du  =  -   --dx  +  —  —  dy.     Q.  E.  D. 
rfz  dx  dz  dy 

SCH.  —  The  student  should  not  fail  to  observe  that  all  the  truths  developed 
in  this  section  are  but  deductions  from  the  proposition  that  the  total  differ 
ential  of  a  function  of  several  variables  equals  the  sum  of  the  partial  dif 
ferentials.  With  this  key  in  hand,  he  can  readily  unlock  the  mysteries  of 
the  whole  subject. 


SECTION   VI. 

Successive  Differentiation  of  Functions  of  Two  Independent 
Variables,  and  of  Implicit  Functions, 

106.  Prop. — Iti  a  function  of  two  independent  variables,  both  va 
riable*  may  be  considered  equicrescent ;  i.  e.,  their  differentials  may  be 
regarded  as  constant. 

Ii'L- — This  proposition  is  an  axiom,  and  it  is  only  necessary  that  its  import  be 
clearly  understood.  Thus,  if  u  =f(x,  y)  and  x  and  y  are  independent,  any  change 
which  x  may  undergo  does  not  affect  y,  and  any  change  which  y  may  undergo 


54  THE  DIFFERENTIAL   CALCULUS. 

does  not  affect  x,  as  this  is  what  is  meant  by  their  being  independent.  We  may 
therefore  conceive  eadi  of  them  to  change  according  to  any  law  we  please  ;  and  it 
is  found  convenient  to  conceive  that  x  increases  by  equal  infinitesimal  increments, 
as  heretofore,  and  that  y  also  increases  by  equal  infinitesimal  increments.  Thus 
dx,  and  dy  are  constants  ;  but  it  does  not  follow  that  we  are  to  regard  dy  =  d.v. 
In  fact  this  would  be  to  establish  a  relation  between  a;  and  y,  and  hence  would  be 
contrary  to  the  hypothesis. 

107.  DEF.  —  When  u  =  f(x,  y),  and  x  and  y  are  independent  of 
each  other,  dfu  and   d,,u  are,  in  general,  functions  of  x  and  y,  and 
hence  may  be  differentiated  with  respect  to  either  variable,  thus  ob 
taining  a  class  of  Second  Partial  Differentials.     In  lite 
manner  these  second  partial  differentials  are  in  general  functions  of 
x  and  y,  and  may  be  differentiated  with  reference  to  either,  giving 
rise  to  Third  Partial  Differentials  ;  etc.,  etc. 

108.  Notation.—  Having  u  =  f(x,  y),       /te*,  dxdy  or 


'*U  dydx,  and  —  dy2  are  the  symbols  for  the  second  partial  differen- 
dijdx  *  dif  J 

d3u 
tials.      The   third   partial  differentials   are   indicated   thus,    -5—  ;^3> 

d*u    ,  d*u    .  d*u    .  d3u    .     .  d*u 

-  —  —dx*dy  or  --r—dydx*t  .  ,  dxdy*  or  -r-~r-dy*dxi  and  -r-dy3. 
dx*dy  dydx*  *  dxdy*  dy'-dx  dy* 

In  each  case  the  form  of  the  numerator  indicates  the  number  of  dif 
ferentiations,  and  the  denominator  the  variable  or  variables  with 
which  the  successive  differentiations  have  been  made,  and  the  order. 

Thus  -^—dy^dx  signifies  that  u  =  f(x,  y)  has  been  differentiated 
dy*dx  y 

three  times  in  succession,  twice  with  reference  to  y  and  once  with  re 
ference  to  x,  and  in  this  order.  So,  in  general,  -.^  -dx^dy*  sig 


nifies  that  u  =/(#,  y)  has  been  differentiated  m  —  n  times  in  succes 
sion  with  reference  to  x,  and  then  n  times  with  reference  to  y. 

Ex.  1.  Given  u  =  x*y*  to  form  the  several  successive  partial  dif 
ferentials. 

du  du  d~u  ~     j 

Results,  — dx    —    %y2xdx,    -rdy    =    2x*ydy ;     T",^2       :   *y*d&, 

d2u  d-u  d3u  _ 

dxdv    =    hxvdxdy :       r—dy2    =    2x*dy* ;      j-«^3    =    ^> 
dy*  ax3 

d*u  d3u  A 

dxdy*  dy3 

— ~     ~     ^^  Q.CIX  dy  ,  ecc. 
dx2dy3 


SUG'S. — Having  -^-dx  =  ty^xdx,  to  differentiate  it  with  respect  to  x,  we  notice 

/du    \ 

in  the  first  member  that  it  will  give  — dx  ;  the  dx  being  written  in  the  de 
nominator,  and  as  a  factor  to  designate  with  respect  to  which  variable  the  differen 
tiation  is  made,  and  also  in  accordance  with  the  principle  that  the  differentialr 
coefficient  multiplied  by  the  differential  of  the  variable,  is  the  differential  of  the 
function.  Now,  the  differential  coefficient  being  the  differential  of  the  function 
divided  by  the  differential  of  the  variable,  we  have  for  the  differential  coefficient 


*. 

of  --d#  taken  with  respect  to  x  —  ;  hence  the  differential  is  dx. 

dx  dx  dx 

d2w 
Finally,  observing  that  da;  is  constant,  this  becomes  — - — dx,  or  -pdx~-     In  h'ke 

manner  the  student  should  analyze  the  other  processes.  It  is  of  the  utmost  impor 
tance  that  he  fully  comprehend  the  reasons  for  these  processes ;  if  he  do  not  he  will 
become  hopelessly  entangled  in  the  subsequent  operations. 

Ex.  2.  Produce  the  successive  partial  differentials  of  u  =  (a;  -J-  y}m 
with  respect  to  x  ;  also  with  respect  to  y. 

Results,  dxu  or  -j-dx  —  m(x  -f  y}m~ldx, 
dx 

j^-dx\  or  dxdxu  =  m(m  —  l)(x  +  y^dx*, 

-j—dx3  or  dxdxdxu  =  m(m  —  l)(m  —  2)(x  -f  y)m~sdx3,  etc.  * 

Ct/tJC 

Sen.— When  u  =f(x,  y],  the  following  forms  are  called  Partial  Differ- 

d*u    d2u     d*u     (Pu      d*u        d*u  dmu 

enttal  Coefficients :   •— .   -7—,  - — — ,  -r-,    ,     ,  ,    ,    .    . -— - — 7—, 

dx2    dy*    dydx    dx'3    dx2dy    dxdy*  dxm~"dyn 

etc. 

Ex.  3.  Form  the  successive  partial  differential  coefficients  of  u  = 
sin  (x  -f  y)  with  respect  to  y. 

Results,  ^  =  cos(>  +  y),  —  ==  —  dn(a? -f  y),  ~  =  —  cos(o;  -f  y)y 

d*u  .    d5u 

—  =  sm  (x  +  y\  -j-±  =  cos(^  +  y),  etc. 

Ex.  4.  Form  the  successive  partial  differential  coefficients  of  u  = 
cos(x  • —  y)  with  respect  to  x. 

du  d*u  .  dau        .    .          . 

Results,  —  =  —sm(^— ?/),  —  =  _cos(«— y),  —  ==sm(a?— y), 

etc. 
Ex.  5.  Form  the  successive  partial  differential  coefficients  of  u  = 


56  THE  DIFFERENTIAL  CALCULUS. 

log  (x  +  y)  with  respect  to  x,  and  also  with  respect  to  y  in  the  com 
mon  system  of  logarithms. 

du  m        d*u  m          d*u         2m(.r  -f-  y) 

.  ~ —  — • '-"•-  *    ~~^ —  •-' "•  —  ~~.    — — — — -      — -      —  "-•  • 

dx         x  -f  y    dx*  (x  -f-  y}z    dx3  (x  -f  y)4 

,  etc.     The  partial  differential  coefficients  with  respect 


(x  + 

to  y  are  altogether  similar. 


109.  Prop. — y  u  =  f(x,  y),  in  which  x  and  y  are  independent, 
and  several  differentiations  be  performed,  m  with  reference  to  one  variable 
and  n  ivilh  reference  to  the  other,  the  result  is  the  same  whatever  the  order 
of  the  operations. 

DEM.— 1st.,  To  show  that  -——dxdy  =  -—rdydx. 
dxdy  dydx 

~dx  =f(x  -f-  dx,  y)  —f(x,  y),  and 
d*Udxdy  =  f(x  +  dx,  y  +  dy)  —  f(x,  y  -f  dy)  —  tf(x  +  dx,  y)  —f(x,  t/)]   = 


fady 

f(x  +  dx,  y  +  dy)  ~f(x,  y  +  dy)  —f(x  +  dx,  y\  +  /(x,  y). 

Again,  ~-dy  =  /(x,  y  -f  dy)  —  /(x,  y)  and 

=  f(x  +  dx,  y  -f  dy)  —  /(SB  +  da;,  y)  —  [/(.r,  y  -\-  dy)  —  f(x,  y)}  = 
f(x  +  dx,  y  +  dy)  —/(a;  +  dx:  y)  —  /(x,  y  +  dy)  -f  /(«,  y). 
These  two  results  being    identical,  we  have  ——-dxdy  =          dydx,  (1). 

2nd.,   To  show  that  -^-  dx2dy  = 


For  convenience  of  notation  put  dxdyu  for  -—  :-  dxdy,  and  dydxw  for  -—  -  dydx. 

dxdy  dydx  * 

Then,  as  before  shown  dvdxu  =  dxduu  =  f(x,  y),  and  hence  may  be  differentiated 
with  reference  to  x  or  y.     Differentiating  with  reference  to  x,  we  have,  dxdydxu  = 


But  by  (1)  d.d^w)  =  dud£(dxu)  or  d.d.dxM.     Whence 


In  like  manner  we  may  proceed  to  any  extent  desired. 

d5?/  ° 

[Let  the  student  show  that  -  —  dx3dy2 
32 


Ex.  1.  Given  u  =  xlogy  in  which  x  and  y  are  independent,  to 
form  the  several  second  and  third  partial  differentials,  and  to  show 
that  dvdxii  =  dxdyu,  and  also  that  dvdxdxu  =  dxdgdxu  =  dxdxdvu.* 


Remits,        dx*  (i.e.  dxdxu)  =  0,         rdxdy  (i.e.  dxd,u)  =  —, 


SUCCESSIVE  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  VARIABLES.   57 

d-u        >  dydx   d*u  _  xdip      d3u 

and  -—-dydx  also  =   -2 

'  y 


dxdy*  d3u 

—Z-,  and  •  •  _   •  ==  0. 
yz  dydx2 

Ex.  2.  Given  u  =  x3y  -f-  ayz,  to  form  the  second,  third,  and  fourth 
partial  differential  coefficients,  and  show  the  convertibility  of  the  in 
dependent  differentiations. 

du  du  d2u  d2u  d*u 

Besutts,  j-  =  ^y,  -  =  x*  +  2ay;  — •  =  Gxy,  --  =  2a,  —- -  =  3^ 


,    d*u  d3u  d3u  d3u  d3u 

and  -— —  =  3^2  also  ;  —  =  6y,  -r—  =  0,  -—  — =  6a?  =  ,    ,    , 
dydx  '  dx*          *  dy3         '  dx*dy  dydx* 

d3u  d3u        d4u 

- — --  ==  0  =  -r— r~  ;    3—  =  0,  etc.,  etc. 
dxdy*  dy*dx     dx* 

Ex.  3  to  7.  As  above  form  and  compare  the  successive  partial  dif 
ferential  coefficients  of  the  following:  :   u  = —  •   u  =  tan"1-  : 

&  +  y*  y 

u  =  sin  x  cos  y  ;  u  —  xv ;  and  u  =  (x  -f  y)4- 


110.  Prob.  —  To  form  the   successive   differentials  of  a  function 
of  two  independent  variables. 

DEM.  —Let  u  =f(x,  y),  in  which  x  and  y  are  independent  variables.     The  total 
differential  being  equal  to  the  sum  of  the  partials,  we  have 

*-£*  +  «*  » 

Now,  remembering  that  as  x  and  y  are  independent  and  hence  may  be  treated  as 
equicrescent,  dx  and  dy  may  be  considered  constant,  and  remembering  also  that 

—  ,  and  —  are,  in  general,  functions  of  x  and  y  (lOf),  we  proceed  to  differentiate 
(1)  again,     Thus 


\dx/       dx2           dxdy 

,/dw\         d2u  ,          d% 
and                                          d(  3-  )  =  ^-1-^*  +  T~  dy, 
\dy/       dydx           dy* 

which  substituted  give 

t  +  2p?-dxdy  +  d~dy*, 
dxdy              dy*  9 

dx*         '   d^-dy               dydx           '   dy*  y        dx* 

since  ^-—-dxdu  =  -——dydx  (1O9}. 
dxdy                ayax 

Again,  differentiating  this  second  differential,  we  have 

du 
*  To  perform  this  operation  observe  that  —  is  treated  as  a  function  of  x  and  y,  and  hence  ita 

total    differential  is  equal  to  the  sum  of  its  partial  differentials.      The  partial  differential  with 
respect  to  x  is  d^dx,  and  with  reference  to  y, 


58  THE   DIFFERENTIAL   CALCULUS. 


7/d%\       d%  ,          d:?w 

but  d(  —  -  )  =  —  -dx  H  --  dw, 

VdxV       dx:>        r  dar-'di/ 


\dxdy 

,/d2M\  - 

<  *  ^  +  '          Substituting,  we  have 


since  -       -dx2dy  =  -7-— — —  dxdydx  =  -   ,    dydx2,  etc. 
d&dy  dxdydx  dydx'2  J 

In  like  manner  we  may  proceed  to  differentiate  as  often  as  desired. 

Son. — A  little  observation  will  enable  the  student  to  write  out  any  re 
quired  differential  of  u  =  f(x,  y}  by  analogy  from  the  above.  He  only 
needs  to  notice  that  every  distinct  form  of  the  partial  differential  of  the 
required  order  is  invoked,  and  making  x  the  leading  letter  insert  the  coef 
ficients  as  in  the  binomial  formula.  Thus  d'au  =  ~dx'3  -4-  5——dx4dv  4- 

dx*  dx*dy 


cbfidy*  dx*cly*  dxdy*  •     y    ~  dy 


[.  frob. — To  form  the  successive  differential  coefficients  of  an 
implicit  function  of  a  single  variable. 

SOLUTION.— Let  u  =f(x,  y}  =  0,  in  which  y  is  an  implicit  function.     We  are  to 

d'y  d% 

form  — -.  —Z,  etc. 
dx2  dx3 

du 

First  we  have  -^  == by  (W2).     (1). 

dx  du    " 

dy 
The  form  for  differentiating  this  is 

<du\du         /du\du 
"dxJdy  ~    Vdv/dx    .     , 

~T~,  =  —  ,  —  dx,  since  the  second  member  is  a  fraction. 

dx2  /du\2 

(dy) 

To  perform  the  operations  thus  indicated  we  have  to  remember  that  —  and  — 

dx          dy 

are  functions  of  x  and  y. 

^          ,/du\       d"u ,  d':u 

Hence  d(  —  )  =  —dx  4- 
\dx/       dx2 


SUCCESSIVE  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  VARIABLES.    59 

and  d(  -- )  =  ——dx  4-  -r-dy.     Dividing  these  values  by  da;  and  substituting, 
\dy/        dydx        }    dy< 

*Y?!f  4.    d"M  -  -¥\  —  --(  d'U    4-  ^  -¥\ 
d2?/  dyVdx2       d-rd?/  dx/        dx\dydx       dy-  dx) 

we  have  -^  =  —  7  .  .. 

dx2  /du\2 

(dy) 

Finally,  substituting  in  this  the  value  of  ~  as  given  in  (1)  we  have 

/  *«\        /  *\ 

du  I  d-u          d-u    dx  I          du  I    d'-u          d2w  da*  I 

dy  I  dx2        dx  dy  du  I          dx  I  dy  dx        dy*  dri  I 

d*y  V  ~W  V  <fy/ 


dy/ 

dw\2/d"u\          dnw    du  du          d"u    du  du        d'2u/du\2 
dy/  \dx2/         dx  (/?/  r/,r  d?/        dydx  dx  fly        ihr\<lx) 

f^lV 

\dy/ 

^^Y  _  2 d ^  --  -  +  —  (— Y 

dxdy  dx  dy        dy'2\dx/ 


In  h'ke  manner  the  higher  coefficients  may  be  produced,  but  the  forms  are  too 
complicated  for  elementary  purposes. 

Ex.  1.  Form  the  first  and  second  differential  coefficients  of  y  as  a 
function  of  x,  when  ?/2  —  laxy  +  ^'2  —  6*  =  0. 

dw 

dy      dx      —  2«v  4-  2.r    «?/  —  x 

SOLUTION.  --  =  —  —  =  --  -  —  '  --  =  —  -  .   For  convenience  of 
dx      du       2y  —  '2ax     y  —  ax 

dy 

dy      ay—x  T^P"! 

notation  put  --  =  —  --  =  p,  whence  p  is  a  function  of  x  and  y.     Hence     --  = 
v     dx       y—ax  LdxJ 

*  But    r*?l   =  *?,    **  = 

' 


.  , 

dx  dy    dx  LdxJ          da;*'     da;  Vy  — 

—  (y  —  aar)  +  a(ay  —  %)       A  dp*  _  -,  fay  —  x\    .         __  a(y  —  aar^  —  (ay  —  a) 

--  —  -  .   clUCl  --     -  Cvu\    -   )  -  (*V  —  •  t  -  * 

(y  —  ax)*  dy  \y  —  ax/  (y  —  ax)2 

Eeducin»   --  =  —  ~.  and  —  =  ---  -.     Substituting  these  values  and 

D>  dx       (y  —  ox)2          dy  (y  —  ax)* 

also  the  value  of  --  as  at  first  found,  we  have, 
dx 

d?y  _  (a2  —  Dy        (a2  —  Ite       ay  —  x  _  (a2  —  l)(?yg  —  2«xy  -f  x2) 
dx2  ~~  (y  —  ax)2        (y  —  ax)2       y  —  ax  ~  (y  —  ax)* 

Ex.  2.  Form  the  first  and  second  differential  coefficients  of  y  as  a 
function  of  or,  when  y2  -f-  #2  —  rj  =  0. 

dy  #  c?2?/  r* 

Results,  -/-  =  —  -,  -5—  =  --  . 
5  6to  y'  da?*  y3 

*  Remember  that  these  are  partial  differential  coefficients. 


60  THE  DIFFERENTIAL  CALCULUS. 

SUG.  —  Be  particular  to  use  the  method  now  being  illustrated. 

Ex.  3.  Given  &  +  Zaxy  +  y3  =  Q,  to  form  ,  and  ,  by  the  method 
for  differentiating  implicit  functions. 

Results,  d/  =  -  X-l,  _ 

ax  ?/2  -f-  ax  ax2        (y*  -f  ax)* 

Ex.  4.  Given  y*  —  2xy  4  a2  =  0,  to  form  the  first  and  second  differ 
ential  coefficients  of  y  as  a  function  of  x,  by  substituting  in  (1)  and 
(2)  of  the  preceding  demonstration. 

SDG's.     ^  =  _2,,,lU  =  22/-2x,^  =  0,   ^  =  -2,^  =  2.       •    dJ- 
dx  1    dy  dx*  dxdy  dy*  dx 

<Pu/du\2         d*u   dudu      d*u/du\* 

-2y      ==       y         and    d?y  _  _  da-Ad?//   ~    d^ty  dxdy^~  dy\dx) 
2y  —  2x         y  —  x  dx2 


/cfoiy 
\dy) 


—  2s) 


Ex.  5.  Given  cos  (x  +  y)  =  0,  to  form  -J--,  and  ~2  by  substituting 
as  above. 

q     '       du  _  du  _  cZ2w  (?% 

dx  'dy  '  dx2  ~  T-JMi  ^^  — 

—  cos  (x  4-  2/)j    i —  —  —  cos  (x  ~\-  y)'      Substituting,    --  =  —  1     and    —    = 
dy2  dx  c/.c- 

—  cos  (x  4  ?/)  sin2(a;  47/^4-2  cos  (x  +  ?/)  sin2(cc  4-  y}  —  cos(x  +  y}  sin2(x  4- y) 

—  8in3^4-  y~ 

These  results  are  as  might  have  been  anticipated,  since  for  cos  (x  4-  2/)  =  0, 
a;  4-  y  =  90°  ;  hence  as  one  arc  (x)  increases,  the  other  (y)  decreases  at  the  same 

rate.     Therefore  —  =  —  1,  and,  consequently,  — -  =  0. 
dx  dx2 

Ex.  6.  Solve  Ex's  1 — 3  inclusive  by  substituting  in  the  general/orw- 
ulce  (1)  and  (2). 


DERIVED   EQUATIONS. 

1 12*  From  u  =  0  =/(#,  y),  we  have 

du 

dy  <te 

dx~       ~  du?     (L)> 
dy 


CHANGE   OF  THE   INDEPENDENT   VARIABLE.  61 

du/d*u        d*u   dy\        du/  d*u  d~u  dy\ 

d!y  dy\dx*       dxdy  dx/        dx\dydx  dy~  dx' 

d&  =  fdu^*  -'     (2)' 

(dy) 

From  (1),  we  have  - — r^  +  —  =  0.     (1,),  which  is  called  The 
dy  dx        dx 

First  Derived  Eqiiation,  or  The  Differential  Equa 
tion  of  the  First  Order. 

From  (2)  we  obtain 


du/d*u        d*u   dy\        du/  d*u        d^u  dy\ 
d\dx2      dxd    dx)       dx\ddx       d2  dx/ 


du  dzy  dy\dx2      dxdy  dx         dx\dydx       dy2 

dy  dx*  du 


du 


dy      dx/  d*u        d*u  dy\ 
dx       du\ddx       d*  dx/ 


dx*       dxdy  dx       du\dydx       dy* 

dy 

d*u  d*u  dy  dy/  d*u  d*u  dy\ 
dx*  dxdy  dx  dx\dydx  dy*  dx/ 
d*u  ^  d*u  dy  d*u/dv\* 

_ O „__.-_  _*-_^ I   */i   l 

dx*         dxdy  dx       dy\dxs  ' 
Whence,  transposing,  we  have, 

y_     l     O  •'     I /_£.  J     J Q         /O  N 

dy  dx*         dxdy  dx       dy^dx/     '   dx* 

which  is  called  The  Second  Derived  Equation  or  TJie 
Differential  Equation  of  the  Second  Order. 

In  a  similar  manner  the  Third  Derived  Equation  is  found  to  be 

dy  dx3  ( dxdy        dy*  dx )  dx*        dy*\dx)  dxdy*\dx/ 

3  &u    dy^        <ftu  _  Q 
dx*dy  dx        dx3 


SECTION  VII. 
Change  of  the  Independent  Variable, 

113.  In  considering  functions  of  a  single  variable,  as  y  =  /"(#), 
the  hypothesis  which  we  usually  make  that  x  is  equicrescent,  and 
hence  that  dx  is  constant,  gives  to  all  the  differentials  and  differential 
coefficients  of  the  function  after  the  first,  a  different  form  from  what 
they  would  have  had  if  such  hypothesis  had  not  been  made.  Thus 


62  THE   DIFFERENTIAL     CALCULUS. 

,/dy\       d-y  d^i/dx  —  d-xdy 

d{  --  )  =  —  -,  when  x  is  equicrescent,  but  —  -  -  -  --  -.  when  neither 

\dx<       dx  dx* 

variable  is  regarded  as  equicrescent  (i.  e.  when  dy  and  dx  are  both 
treated  as  variable).  In  the  course  of  a  discussion  it  sometimes  be 
comes  important  to  change  the  conception  and  regard  y  as  the  equi 
crescent,  or  independent  variable,  and  x  as  the  function.  Or  it  may 
be  desirable  to  introduce  a  new  variable  of  which  x  is  a  function,  and 
make  it  the  equicrescent  variable. 

Either  of  these  changes  can  be  readily  effected  by  first  giving  to 
the  expression  under  consideration  the  form  which  it  would  have  had 
if  neither  variable  had  been  treated  as  equicrescent.  Then,  to  make 
y  equicrescent,  remember  that  all  its  differentials  above  the  first  are  0, 
and  drop  out  the  terms  affected  by  them.  To  introduce  a  new  inde 
pendent  equicrescent  variable,  as  0,  of  which  x  is  a  function,  simply 
substitute  in  the  general  form  in  which  neither  x  or  y  are  equicres 
cent,  the  values  of  x,  dx,  d-x>  etc.,  in  terms  of  the  new  equicrescent 
variable  0. 

114.  Prob.—Tofind  the  forms  which  -^,  ^~,  ^  etc.,  take  when 
neither  variable  is  considered  equicrescent. 

DEM.  —  Since  the  hypothesis  of  an  equicrescent  variable  has  not  modified  the 
form  of  --,  in  it  x  or  y  may  be  considered  equicrescent,  or  neither,  at  pleasure. 

d(^-\ 

Again  -^  =     ^  X   .     Now  differentiating  the  latter  without  regarding  dx  as 
dx*  dx 


/  dy  \        fcy  dx  —  d^x  dy 


constant  We  have     &    =  =      £=  which  is  therefore 

dx  dx  dx* 

the  form  which  the  second  differential  coefficient  takes  when  neither  variable  is 
equicrescent. 

j(d<ly\  jS&yfa    --    d*xdy\ 

d*y  \dx-V  \  &*  ' 

Once    more,   -£    =    -^5  —  -    =    -  1  -    = 
dx3  dx  dx 

—  d3x  dy  dx3  —  d* 


(dtydx  - 


the  third  differential  coefficient  when  neither  variable  is  equicrescent. 

Ex.  1.  Transform  x—  -1-  (—  V  —  -^  =  0  in  which  x  is  equicres- 
dx*       \dxs         dx 

cent,  into  its  equivalent  when  y  is  equicrescent. 

d*y       d*y  dx  —  d^xdy  __ 
SOLUTION.  —  When  y  is  equicrescent  d*y  ==.0,  hence  —  =         ~~7^  — 


CHANGE   OF  THE  INDEPENDENT  VARIABLE.  63 


Substituting,  we  have  -  x  +  ^  -  |  =  0.     Dividing  by  dif 

and  multiplying  by  dx3  to  give  the  differential  of  the  independent  variable  its 

d^x       /d.v\2       1        A 
proper  position,  and  changing  signs,  we  obtain,  #—  +  (  --  j  - 

Ex.  2.  Transform  (dyn>  +  dx*)1*  +  adxd*y  =  0,  in  which  x  is  equi- 
crescent,  into  its  equivalent  when  y  is  equicrescent. 


Ex.  3.  Transform  -~  —  = ;  -j-  +  ^ — : — -  =  0,  in  which  x  is 

equicrescent,  into  its  equivalent  in  terms  of  0  as  the  equicrescent  va 
riable,  when  x  =  cos  6. 

SOLUTION. — Introducing  the  general  form  of  the  second  differential  coefficient, 

d2v  dx  —  d*x  dy           x      dy           y 
the  given  equation  becomes,-— — —         —  ^  _  ^  ^-  -f-  ^ ^  =  U- 

Now,  from  x  =  cos  0,  dx  =  —  sin  0  d0,  d*x  ==  —  cos  0  d02,  and  1  —  .r2  =  sin2  0. 
ing  these  values,  we  have,  — 

0,  or  reducing,  -^  +  y  =  0. 


—  rZ2?/  sin  0  dO  -f-  cos  5  rfi9^  dy    .     cos  Q       r/?/ 
Substituting  these  values,  we  have,  -  :-_____  --  +  ^-^  —  -~g  4. 


siu 


\  CA  X     * 

Ex.  4.  Transform  R  =  ,  in  which  x  is  equicrescent,  into 

dx* 

its  equivalent  in  terms  of  the  variables  r  and  0,  0  being  the  equicres 
cent  variable,  when  y  =  r  sin  0,  and  #  =  r  cos  0. 

SUG'S. — The  formula  in  its  more  general  form,  in  which  neither  variable  is  re 
garded  as  equicrescent  is 


n 


From  y  =  r  sin  0,  we  have,  dy  =  sin  0  dr  +  r  cos  0  d0,  and  <f?7/  =  sin  0  d-r  -f- 
2  cos  0  dQ  dr  —  r  sin  0  c?02.  From  a;  =  r  cos  0,  we  have,  dx  =  cos  0  dr  —  r  sin  0  d0, 
and  d7x  —  cos  0  d2r  —  2  sin  0  d0  dr  —  r  cos  0  d02.  Substituting  these  values  and 
reducing,  we  have, 


.  The  method  just  given  is  sufficient  to  resolve  all  cases  of 


64  THE   DIFFERENTIAL   CALCULUS. 

change  of  the  equicrescent  variable,  but  general  formulas  are  some 
times  convenient.     We  proceed  to  deduce  the  most  important. 

HO.  J*vop.  —  In  operations  where  y  has  been  treated  as  a  function 
of  the  equicrescent  variable  x,  to  change  the  conception  so  that  x  shall  be  a 

dv    1 

function  of  the  equicrescent  variable  y,  we  substitute  for  -^-,  -—  and  for 


d2y        dy2 
dx*'        dx3' 


DEM.  —  As  the  hypothesis  of  the  equicrescent  variable  does  not  affect  the  first 

differential,  we  have  the  identical  relation  --  =  —  . 

dx      dx 

dy 

,          d*y       d*yda:  —  d^xdy 
Again,  if  neither  variable  is  equicrescent  we  have    —  =  —  -  --  —  -  --  -. 


Now,  making  y  equicrescent  makes  d^y  =  0  ;  hence  -^  =  —  •  —  —  -  =  —  -^-. 

dx*  ax3  ax3 

<ty3 

Q.  E.  D. 

Sen.  —  In  a  similar  manner  the  corresponding  forms  for  the  higher  coef 
ficients  can  be  deduced  ;  but  they  are  not  often  required. 


.  J*vop.  —  In  expressions  where  y  has  been  treated  as  a  function 
of  the  equicrescent  variable  x,  to  change  the  expression  so  that  y  shall  be  a 
function  of  some  new  equicrescent  variable  as  0,  having  given  x  =  <p(0), 
dy  d2y  dx       d2x  dy 

dy      dO  d«y      di?*  30  ~~  30*  30 

we  substitute  for  ~t     •=-,  and  for  —  -,     -  -  - 
dx      dx  dx2  clx3 

cO?  303 

DEM.—  Since  y  =  f(x)  and  x  =  <p(&)t  we  have  (104),  -~  —  ™  ^  ;  whence 

dy 

dy  _  dB 

dx  ~~~  dx 
dQ 

cfiii 
Again,  the  general  value  of  -=-£  (when  neither  variable  is  treated  as  equi 

crescent)  is 


d*y  _  d^y  dx  —  d?x  dy 
dx2  ~  dx* 

dx 
Now  from  x  =  q>(Q)  we  have  dx  =  -^dO.     Differentiating  this,  remembering  that 


CHANGE   OF  THE   INDEPENDENT   VARIABLE.  65 

dQ  is  constant,  we  have,  &x  =  jjdO*.      Substituting    these   values  of   dx   and 

dta,    in    the     general    value    above,    we  obtain     ^    =    d'y  dx 

dx*  d 

&yte__#*dy 

dV  dQ       dQ*  dQ 


Sen.—  To  apply  these  formula*  in  practice,  we  make  the  requisite  substi- 
dy  d*y  dx       d*x  dy 

tutions  of  ~  for  ^,  and  -     ^  *  —  for  ^  in  the  given  expression, 
dQ  d& 

and  then  finding  the  value  of  —  and  of  ~  from  the  relation  x  =  G>(6), 

dQ  dQ*  . 

substitute  these  values,  and  have  an  expression  in  terms  of  -£  and  —  ^  as 

dQ  dQ* 

required. 


Ex.  1.  Given  —  =  ^  in  which  x  is  the  equicrescent  (in 

dependent)  variable,  to  transform  so  that  t  shall  be  the  equicrescent 

variable,  knowing  that  x  =  log  —  . 

\/l  —  t* 

SUG'S.—  Substituting  the  value  of  -^  requisite  for  this  transformation,  we  have 

d?y  dx  _  d^x  dy 

dP  dl~di?   dt 


(ex  +  e~xY 


troducing  these  values,  the  expression  becomes  —  -  ^  __  f-     1  ~ 

dt*     t(l  —  <«)    T    <2(1  _ 


o«-^)~+(i-3^ 

by  observing  that  &  =        l       ,  and  er-*  =  ^~*  ;   whence   (^  +  e-«)s  = 
_l  _ 

BgTT^ 

Ex.  2.  Transform  ^  +  ^  +  y  =  0,  into  a  form  in  which  *  is 
the  equicrescent  (independent)  variable,  knowing  that  x*  =  U. 


66  THE  DIFFERENTIAL  CALCULUS. 

Sen. — We  observe  from  the  foregoing  that  a  change  of  the  equicrescent 
variable  may  greatly  simplify  the  expression. 

Ex.  3.  Making  x  =  cos  t,  and  t  the  equicrescent  variable,  show  that 

(1  _  ^p.  -  o^  =  0,  becomes  ^  =  0. 

' dx*         dx  dt* 


118.  Prob. — Having  u  =  f(x,  y),  where  x  =  cp(r,  0)  and  y  = 
<pi(rtO)tto  express  the  partial  differential  coefficients  —  and  —  -  in  terms 

of  r,  6  and  the  partial  differential  coefficients  — ,  and  -r-. 

SOLUTION. — Since  u  is  a  function  of  x  and  y,  each  of  which  is  a  function  of  r, 
we  have 


And  in  like  manner 


du  du  dx  du  dy 

dr        dx  dr  dy  dr' 

du        du  dx  du  dy 

d0  =  dx  dO  dy  dO' 


From  these  equations  eliminating  --,  and  finding  the  value  of  — ,  we  obtain 

du  dy        du  dy  du  dx        du  dx 

du        dr  dO  ~~  dQ  dr  du  dr  dQ  ~~  dO  dr 

—  =  - — • • — — ,  and  similarly,  —  = ; — • — — . 

dx        dx  dy        dy  dx  dy  dx  dy        dy  dx 

drdQ~drdQ  dr  dQ  ~  dr  dB 

119.  COR. — If  x  =  r  cos  0,  and  y  =  r  sintf,  the  above  formula?  be- 

du  du       sin  0  du        _  du  n  du       cos  0  du 

come  —  =  cos  0 -=-,  and  —  =  sin  0  - — | — . 

dx  dr          r     do  dy  dr          r     d0 

Sen.  1. — It  will  be  observed  that  the  relations  x  =  rcosO  and  y  =  rsmO, 
are  the  common  formulas,  for  passing  from  rectangular  to  polar  co-ordinates 
(PART  L,  128). 


GENERAL   SCHOLIUM, 

We  here  conclude  the  subject  of  the  Differential  Calculus,  having  de 
veloped  the  theory  as  fully  as  the  plan  of  our  course  requires.  In  the  next 
chapter  we  shall  give  a  few  applications  ;  but  its  transcendent  efficiency  is 
best  seen  in  the  General  Geometry  and  in  Physics. 


CHAPTER  IL 

APPLICATIONS  OF  THE  DIFFERENTIAL  CALCULUS. 


SECTION  Z 

Development  of  Functions, 

120.  DEF.  —  A  Function  is  said  to  be  Developed  when  the  in 
dicated   operations   are   performed  ;   or,  more  properly,  when  it  is 
transformed  into  an  equivalent  series  of  terms  following  some  gen 
eral  law. 

ILL'S.  —  Division  affords  a  method  of  developing  some  forms  of  functions.  Thus 
y  —  -  -  when  developed  by  division  becomes  y  =  1  -J-  x  -{-  x-  -f-  x3  -f-  ,  etc. 

The  binomial  formula  (COMPLETE  SCHOOL  ALGEBRA,  195}  is  a  formula  for  develop 
ing  a  binomial.  Thus  y  =  (a  -\-  x)b  when  developed  becomes  y  =  a5  -f-  5a4x  -f- 
10«3#2  +  lOa2^3  4"  5ax*  -f-  x5.  The  subject  is  one  of  great  importance  in  math 
ematics. 

MACLAURIN'S   FORMULA. 

121.  DEF.  —  3£ac&aurin'8  Formula  is  a  formula  for  devel 
oping  a  function  of   a  single   variable   in   terms  of  the   ascending 
powers  of  that  variable  and  finite  coefficients  which  depend  upon  the 
form  of  the  function  and  upon  its  constants. 

122.  J*rol),  —  To  produce  Maclaurin's  Formula. 

SOLUTION.—  Let  y  =  f(x)  be  the  function  to  be  developed.  It  is  proposed  to 
discover  the  law  of  the  development,  when  the  function  can  be  developed  in  the 
form 

y  =  f(x)  =  A  4-  Bx  4-  Cfe*  _|_  J5xs  _|_  fa*  _j_,  etc., 

in  which  A,  3,  C,  D,  etc.  ,  are  independent  of  x  and  depend  upon  the  form  of  the 
function,  and  its  constants. 

Producing  the  successive  differential  coefficients,  remembering  that  A,  B,  C,  D, 
etc.,  are  constant,  we  have, 


-f  3J>a;2  4.  4J5c3  4-,  etc., 


_  2(7  4-  2  .  3JDa;  4-  3  .  4.Ec8  4-,  etc., 

(J3aj 

g  =  2  .  3D  +  2  .  3  •  ±Ex  4-,  etc., 

g  =  2.  3.  4tf+,  etc. 
Now  as  the  coefficients  A,  B,  C,  D,  etc.,  are  independent  of  x,  they  are  the  same 


68  APPLICATIONS   OF  THE  DIFFERENTIAL  CALCULUS. 

for  all  values  of  it,  and  if  we  can  find  what  they  should  be  for  any  one  value  of  x 
we  shall  have  their  values  in  all  cases.  Now,  if  x  =  0  we  have  (y)  =/(0)  =  A, 
the  expressions  (y},  and  /(O)  signifying  the  value  of  the  function  when  x  =  0. 


-«:••  -,  *•  ( 

fying  in  each  case  the  value  of  the  particular  function  when  x  =  0.     Hence  we 


Substituting  these  values,  we  have 


which  is  the  formula  required.  • 

123.  Sen.  1.  —  The  student  should  become  perfectly  familiar  with  this 
important  formula,  and  for  this  purpose  it  will  be  well  to  describe  it  thus  : 
Maclaurin's  Formula  develops  y  =  f(x)  into  a  series  of  terms,  the  first  of 
which  is  the  value  of  the  function  when  x  =  0  ;  the  second  is  the  first  dif 
ferential  coefficient  of  the  function,  x  being  made  0,  into  x  ;  the  third,  the 

second  differential  coefficient,  so  being  made  0,  into  —  ,  etc. 

124.  Sen.  2.  —  This  formula  may  also  be  written   y  =  f(x]  =  /(O)  + 

)  +  /^  +•  etc"  in 


,  /2(0),  etc.,  signify  the  same  as  (y},  ,  ,  etc.,  respectively. 


Ex.  1.  To  develop  y  =  (a  -{-  x)7,  by  Maclaurin's  Formula. 
SOLUTION.  —  Differentiating    successively,    we    have    -£  =  l(a  -\-  cc)fi,    ^-|  = 


—  =  2-3-4.5.6.7(a  +  x\  —  =  1-2-3-4.  5-6-7.     Here  the  differentiation  termi- 
dx<>  dx"1 


nates.     Making  x  =  0,  we  have,  (y)  —  (a  +  0)7  =  a",     --)  =  7(a  +  O)6  =  7a6, 

*»,  4.B.8.70.,          =  8.4.6.6.7«.,         = 


2-  3-4.  5-  6-  7a,  and  =  1.2.3.4.5-6.7. 

\ax7/ 

Substituting  in  the  formula,  we  obtain 
y  =  (a  +  xy  =   en  +  7a««  +  6  -  7a<£  +  5  .  6  .  7^-fl  +  4.5.6.7^^-|L  4. 


or,  reducing,  ?/  =  (a  +  tf)7  =  a^  +  7a6x  +  21a5x2  +  35a«a?  +  SSa-^ar*  +  21a2x5  + 
7ax6  +  x~>,  a  result  identical  with  that  given  by  actual  multiplication,  or  by  the 
Binomial  Formula. 


DEVELOPMENT  OF  FUNCTIONS.  69 

Ex.  2.  To  deduce  the  Binomial  Formula  from  Maclaurin's  Formula. 

SOLUTION.—  Let  y=  (a-\-x)m,  in  which  m  is  either  integral  or  fractional,  positive 
or  negative.     Then  differentiating  successively,  and  taking  the  values  for  x  =  0, 

.«-     (**)    =   m(m  -  1>*~,    (g)    = 


m(m  —  l)(m  —  2)a"1-3,  -7  =  m(m  —  l)(m  —  2)(ra  —  3)am-^  -f  ,  etc.  Sub- 
stittiting  in  Maclaurin's  Formula,  we  obtain  y  =  (a  -\-  x)m  =  am  -j-  mam~lx  -f- 
m(m  —  l)a«-  *J-fm(r»  —  l)(m  —  2)a'»-3^  -f  m(m  —  l)(m  —  2)(m— 


+,  etc.,  or  we  may  write  y  =  (a  -f-  x)m  =  a771  -f  ma"*-1*  +  7^  —  am-2ar2  -f- 

1  '2 

m(m  —  l)(m  —  2)  ,   m(m  —  l)(m  —  2)(m  —  3) 

~  -  —5  --  ^a™-3x3  -f  -2  -  f    —  —  i:  -  -a™-4#4  +,  etc.,  which  is  the 

L-  &-o  l'^S'Q'4 

Binomial  Formula. 

Ex.  3.  Develop  y  =  sin  x. 

Ml  - 


.  1-2.3  ~*~  1.2.3.4.5  ~"  1-2.3.4.5.6.7  +>  etc' 


Ex.  4.   Develop  y  =  cos  or. 

r»       14  x<i  x*  & 

tiesulL  y  =  cos  x  =  1  —  -  -  4-  —  ---  -  4- 

1-2  n    1-2.3-4       1-2-3-4-5-6  H 


1-2-  3-  4-  5-67-8 


>e 


125.  Sen.  —  These  formula  enable  us  to  compute  the  natural  sine  and 
cosine  of  any  arc  directly.     Thus,  to  obtain  the  natural  sine  of  10°,  we  have 

»  =  jg  =  .174533  nearly.     This  value  substituted  in  the  formula,  will  give 

the  sin  10°  =  .17365,  and  cos  10°  =  .98481.     The  series  converge  so  rapidly 
that  but  few  terms  are  necessary. 

Ex.  5.  Develop  y  —  (a2  -f  bx)*  by  Maclaurin's  Formula. 

SUG'S.     y  =  (a~2  +  bx)  ,  .-.  (y)  =  a, 


*  This  notation  signifies  "  x  being  made  equal  to  0." 


70  APPLICATIONS   OF  THE   DIFFERENTIAL   CALCULUS. 

Ex.  6.  Develop  y  =  vl  -+-  x2. 

2t         o       16        liiiS 
etc. 


Ex.  7.  To  produce  the  logarithmic  series. 


SOLUTION.  —  This  series  is  the  development  of  y  =  log  (1  -{-  x).     Differentiatin 
ith  reference  to  a  system  of  logarithms  whose  modulus  is  m,  we  have,   -^ 
m      d?y  _  m         cPy  _       2m         fry  _          2  -  3m 

- 


e 


stituting  in  Maclaurin's  Formula,  we  have 

y  =  log  (1  -[-  a;)  =  m(aj  —  ix2  -f  i.x3  —  1^  -J-,  etc.), 
the  law  of  the  series  being  apparent. 

120.  COE.  1.  —  Since  in  the  Napierian  system  m  =  1,  we  have 
y  =  log(l  +  x)  =  x  —  Jx«  +  i*3  —  Jx<  +  ix*  —  etc. 


.  Sen.  1.  —  This  formula  is  not  adapted  to  the  purpose  of  computing 
logarithms,  since  it  is  diverging  for  integral  values  of  x.  Thus,  letting 
x  =  2,  we  have  y  =  Iog3  =  2  —  2  -f  f  —  4  -f  ^  —  ,  etc.,  in  which 
each  term  after  the  first  two  is  greater  than  the  preceding,  and  hence  ex 
tending  the  series  does  not  approximate  the  value  of  log  3. 

From  the  series  in  the  corollary,  however,  a  converging  series  may  be 
readily  deduced.     The  following  is  a  simple  method  : 

Substituting  —  x  for  x  we  have 

log  (1  —  x}  =  —  x  —  $x*  —  i-r3  —  io*  —  i#5  —  ,  etc. 

Subtracting  this  from  the  former,  we  obtain 


log  (1  +  x)  —  log  (1  —  x)  =  log  =  2(*  +  ^3  +  i*5  +  ^  +,  etc. 

(  1  —  x  ) 

Now  putting  x  —  -  -  -,  whence  -i-2  =  Z—^—,  we  have  log—    - 
2iZ  -|~  1  1  —  x  z  z 

log  (,  +  1)  -log,  =  2  +  .,  +        L_  +  _,+,  etc.  ), 

or  log  (s  +  1)  = 

This  series  converges  for  all  positive  values  of  z,  and  more  rapidly  as  z  in 
creases. 

To  apply  this  formula  in  computing  a  table  of  Napierian  logarithms,  first 
let  z  =  l,  whence  log  2  =  0  -f- 

1  +  J:     +_i_  +  _l_  +  J_  +  _L_  +  -?_  +_!_+.  etc.) 
3  ^  3  •  3».      5  •  3s      7  •  3'  ^  9  •  3»      11  •  3"  T  13  -  31-'  T  16-  8»  / 


DEVELOPMENT   OF  FUNCTIONS. 


71 


The  numerical  operations  are  conveniently  performed  as  follows  : 


3 

2.00000000 

.66666667 
.02469136 
.00164609 
.00013064 
.00001129 
.00000.103 
.00000009 
.00000001 

9 
9 
9 
9 
9 
9 
9 

.66666667 
.07407407 
.00823045 
.00091449 
.00010161 
.00001129 
.00000125 
.00000014 

.'.  log 

1 
3 
5 
7 
9 
11 
13 
15 

'  2  — 

.69314718 

Second. 


To  find  log  3,  make  z  =  2,  whence  log  3  =  log  2  -f 
./l  1  1  1  1 


Third. 
Fourth. 


To  find  log  4. 
To    find    k 


~V5 
5 

1  3-53  '  5 
2.00000000 

5* 

1 

3 
5 

7 
9 

X 
w 

7-5'  '  9.5'  '   *"> 

.40000000 
.00533333 
.00012800 
.00000366 
.00000011 

25 
25 
25 
25 

log  4 
5. 

.40000000 
.01600000 
.00064000 
.00002560 
.00000102 

log  2 
.'.  Iog3 

==21og2  =  2 
Let  z  =  4, 

.40546510 
.69314718 

1.09861228 

69314718  =  1.38629436. 
lence  log  5  =  log  4  -f 

2  ^ 


"9!  +  5—9-5  +  JfTgl  +'  etc</ 


Computation. 


9 

81 
81 
81 

2.00000000 

1 
3 
5 

7 

4  = 

5  = 

.22222222 
.00091449 
.00000677 
.00000006 

.22222222 
.00274348 
.00003387 
.00000042 

log 
.*.  log 

.22314354 
1.38629436 

1  .  60943790 

In  like  manner  we  may  proceed  to  compute  the  logarithms  of  the  prime 
numbers  from  the  formula,  and  obtain  those  of  the  composite  numbers,  on 
the  principle  that  the  logarithm  of  the  product  equals  the  sum  of  the  log 
arithms  of  the  factors. 

The  Napierian  logarithm  of  the  base  of  the  common  system,  10,  = 
log  5  +  log  2  =  2.30258508. 

128.  Con.  2. — The  logarithms  of  the  same  number  in  different  sys 
tems  are  to  each  other  as  the  moduli  of  those  systems  ;  and  the  logarithm 
of  a  number  in  any  system  equals  the  Napierian  logarithm  of  the  same 
number  multiplied  by  the  modulus  of  the  proposed  system. 


120.  SCH.  2. — To  find  the  modulus  of  the  common  system  of  logarithms, 


72  APPLICATIONS  OF  THE  DIFFERENTIAL  CALCULUS. 

we  have  com.  log.  x  —  m  Nap.  log.  x,  whence  m  —  com-   °ff-  ^     jq-QW  jlfty_ 

Nap.  log.  x 
ing  computed  the  Napierian  logarithm  of  10,  by  the  formula  above,  and 

found   it  to   be   2.302585,   we   have   m  =    com-  lo£-  10  =   _  \  _   ___ 

Nap.  log.  10         2.302585 
.43429448+.       . 

13O.  SCH.  3.  —  To  compute  a  table  of  common  logarithms,  first  compute 
the  Napierian  logarithms  and  then  multiply  by  the  modulus  of  the  common 
system,  .43429448. 

Ex.  8.  To  ascertain  the  relation  of  the  modulus  of  a  system  of  log 
arithms  to  its  base. 

SOLUTION.  —  Developing  y  =  ax,  by  Maclaurin's  Formula,  we  have 

„  =  a*  =  1  +  U  +  I  I  +  I  J«L  +  I  _|L_  +  etc.    (1). 
Again,  putting  a  =  1  -j-  6,  and  developing  by  the  Binomial  Formula,  we  obtain 


| 
Expanding  and  collecthig  the  coefficients  of  the  1st  power  of  x  we  find  it  to  be 

.  62  53  54  &5  56 

5-2+r-4+5-6-+'etC- 

Finally,  since  series  (1)  and  (2)  are  equal  the  coefficients  of  like  powers  of  x  are 

1  62       63       64       65       66 

equal,  and  —  =6  —  77  +  5  --  T  ~t~  5  --  c~  ~H  »  e^c<  >  or  restoring  a  and  finding 

771  «  O  &  O  v 

the  value  of  m,  we  have 


(a  —  1)  —  i(a  —  1)2  -f  i(a  —  I)3  —  4(a  —  !)<-}-£(«  —  I)5  —  i(a  —  l)b+,  etc. 


.  SCH.  —  To  find  e,  the  base  of  the  Napierian  system.  Since  the  log 
arithms  of  the  same  number  in  different  systems  are  to  each  other  as  the 
moduli  of  those  systems,  we  have 

com.  loge  :  Nap.  loge(=l)   ::  .43429448  :  1. 

.'.  com.  loge=  .43429448,  and  e  from  the  table  of    common  logarithms, 
which  we  have  shown  how  to  compute,  is  2.718281+. 

Ex.  9.  To  develop  y  =  ax,  i.  e.  to  produce  the  exponential  series. 

x  x~  ^ 

Result,    y  =  a*  =  1  +  log  a  -    +   log2  a—    -f   log3 


.  ^  .  3 


132.  SCH.—  If  a  =  e,  the  Napierian  base,  this  becomes  y  =  e*  =  \-{-- 


X4  X5 

|     etc 

^ 


If  x  =  1,  we  have 


DEVELOPMENT   OF  FUNCTIONS.  73 


23.  2  •  3  •  4  •  5      ' 

finding  the  Napierian  base,  although  the  series  converges  slowly. 


formula  for 


Ex.  10.  Develop  y  =  tan"1^. 

SOLUTION. — Differentiating  ~-  = ,  which  by  division  becomes  --  =  1 

ax       1  +  x2  dx 

:2  +  x<  —  x6  +  x*  —  x><>  +,  etc. 
Differentiating  successively,  — ^  =  —  2x  +  4a;3  —  6x5  +  8x7  —  10x9  +,  etc. 


—  9.10x3+,  etc. 

^  =  2-3-4a;  —  4.5-6x3  +  6-7-8x5 _  8-9-10x7+,  etc. 

d% 

^.  =  2.3.4  — 3.4-5.6x2+5.6-7-8x<— 7-8.9-10x6+,  etc. 

d6?/ 

^  =  — 2-3.4.5-6x+4  5.6-7-8x3  — 6-7-8-9.10x5+etc. 

Whence  W  =  tan--0  =  o,(4)  =  X,  (g)  =  0,  (g)  =  -  2,  (g)  =  0, 

-^j  =  2-3-4,  (-r^J  =  0,   etc.     Introducing  these  values  into  Maclaurin's 
Formula,  we  have  y  =  tan-'#  =  x  —  te3  +  \&  —  \af  +  ^  —  ^a;11  +,  etc. 

Sen. — By  means  of  this  development  we  are  enabled  to  find  the  value 
of  ic.     Thus  let   y  =  4=5°  =  ^,  whence    x  =  I,  and   we  have  y  —  ^  = 

_ni        1,11.1        1         1         1         1         1 
tan  '1  =  1  _-  +  __-  +  ___  +  ___+___+,  etc. 


133.  Prop. — Though  Maclaurin's  Formula  is  applicable  to  a  very 
great  variety  of  forms  of  functions  of  a  single  variable,  it  will  not  develop 
ALL  such  functions. 

The  truth  of  this  theorem  will  be  substantiated  if  we  can  present  examples  of 
functions  of  a  single  variable  which  the  formula  will  not  develop  properly.  This 
we  proceed  to  do. 

Ex.  1.  Show  that  y  =  x2  is  not  properly  developed  by  Maclaurin's 
Formula. 

SOLUTION.— From  y  =  x*,  we  have  ^  —  — ; ,  J^  = -^  etc.     Hence  (y)  =  0, 

2x*  '  4x* 

/dy\       1  /d*y\ 

\dx/  ==  0  ==  °°'  \dx°)  =  —  °V«*°'    Substituting  these  values  in  the  formula  we 

have  y  =  x    =  0  4-  oo  a;  —  oo  —  +,  etc.     Such  results  as  these  will  be  simply 


74  APPLICATIONS   OF  THE   DIFFEKENTIAL   CALCULUS. 

unintelligible  to  the  learner  at  first.     But  in  this  case  it  is  easy  to  see  that  the  de 
velopment  will  consist  of  pairs  of  terms  of  the  same  general  form  as  oo  x  —  <x  —  . 


To  ascertain  just  what  is  to  be  understood  by  this  binomial,  let  us  restore  the 
values  of  oo  as  they  were  before  x  was  made  equal  to  0,  only  using  x   to  indic-aie 


the  x  that  is  0.     We  then  have 


cc 
this  becomes  —  '-  ,  which  is  oo  for  att  values  of  x  except  0,  and  indeterminate  for 

that.     In  like  manner,  it  may  be  shown  that  each  succeeding  pair  of  terms  equals 

oo  .  Hence  we  have  the  absurd  result  that  y  —  x  —  oo,  for  all  values  of  x,  since 
the  development  should  be  true  for  all  values  of  the  variable. 

Ex.  2.  Show  that  y  =  log  x  is  not  properly  developed  by  Maclau- 
rin's  Formula. 

Suo's.  —  The  result  is  similar  to  the  preceding  except  that  the  first  term  is  oo  in 
this  case.  Each  succeeding  binomial  may  be  seen  to  be  oo,  as  in  the  former  case. 
Hence  we  have  the  absurd  result  that  y  =  log  x  =  oo  for  all  values  of  x. 

i 

Ex.  3.  Show  that  y  =  cot  x,  and  y  =  ax  are  not  properly  developed 
by  Maclaurin's  Formula. 

134-.  Sen.  —  The  occasion  of  the  inapplicability  of  Maclaurin's  Formula, 
in  such  cases  as  just  given,  is  the  fact  that  the  form  of  the  function  is  such 

that  the  coefficients  -j-,  —  ^,  etc.,  or  the  function  itself,  or  both,  become 

infinite  for  x  =  0,  which  is  contrary  to  the  hypothesis  upon  which  the 
formula  was  produced.  Whether,  in  such  cases,  the  failure  to  develop  cor 
rectly  by  this  formula  is  due  to  the  fact  that  the  particular  function  is  in 
capable  of  any  development,  or  whether  it  is  simply  because  it  will  not 
develop  in  the  particular  form  assumed  in  this  theorem,  does  not  as  yet  ap 
pear,  and  our  limits  forbid  our  entering  upon  the  question. 


TAYLOR'S   FORMULA. 

135.  DEF. — Taylor9 s  Formula  is  a  formula  for  developing  a 
function  of   the   sum  of  two  variables   in  terms  of   the    ascending 
powers  of  one  of  the  variables,  and  finite  coefficients  which   depend 
upon  the  other  variable,  the  form  of  the  function,  and  its  constants. 

136.  Lemma. — If  u  =  f(x  -f  y)  the  partial  differential  coefficients 

du       _  du 

--  and  —  are  equal. 

dx          dy 

DEM. — Having  u  =  f(x  -f-  y},  if  x  take  an  increment,  we  have  u  -f-  dxu  = 
f(x  +  dx  +  y}  =f[(x  +  y)  +  dx}  ;  whence  dxu  =/[(z  +  y)  -f  dx]  —f(x  -f  y). 
Again,  if  y  take  an  increment,  we  have  u  -}-dvu  =f(x -\-y-\-dy)  —f[(x-\-y)-^-dy]', 


DEVELOPMENT  OF  FUNCTIONS.  75 

whence  dvu  =f[(x  -f-  y)  +  ^/l  —  /(x  4~  2/)-  Now  the  /onn  of  the  values  of  dxu. 
and  tZj/w,  as  regards  the  way  in  which  x  and  y  are  involved,  is  the  same  ;  hence, 
if  it  were  not  for  dc  and  dy,  they  would  be  absolutely  equal.  Passing  to  the  differ 
ential  coefficients  by  dividing  the  first  by  dx  and  the  second  by  dy,  we  have 

du  _/[(*  +  y)  +  <**]  —  /<*  +  */)  and  <fa  =/[fo  +  y  '  +  'fy]  -/^  +  y)     But 

dx  -  dx  dy  dy 

in  differentiating,  the  differential  of  the  variable  enters  into  every  term  ;  hence 
fl(X-\-y)  -f-  dx]  —  f(x  -{-y),  as  it  would  appear  in  application,  would  have  a  dx  in 
each  term  which  would  be  cancelled  by  the  dx  in  the  denominator  in  the  coef 

ficient,  and  —  would  be  independent  of  dx.    In  like  manner  —  is  independent  of  dy. 

Hence,  finally,  as  these  values  of  the  partial  differential  coefficients  are  simply  func 
tions  of  (x  -f-  y)  ,  of  the  same  form,  and  not  involving  dx  or  dy,  they  are  equal.  Q.  E.  D. 

Sen.  —  The  substance  of  this  demonstration  is  that  the  values  of  the  dif 
ferential  coefficients  depend  upon  the  form  of  the  function,  and  are  inde 
pendent  of  the  increment  of  the  variable.  Therefore  when  the  form  of  the 
function  is  such  as  to  give  to  the  partial  differential  coefficients,  the  same 
form  with  respect  to  the  variables,  the  coefficients  are  equal.  But  suppose 


we  have  «=  /fcy).     g  =  /[(*  +  cfafrl  -/fry)  =  /to  +  ycfe)  -  /to)  . 


and  du  -f[x("  +  dy]  ~/kv)  =f(xy  +  Xdy]  ~f(xy\     In  these  coef- 
dy  dy  dy 

ficients  we  see  that  the  form  is  not  such  as  to  involve  x  and  y  in  the  same 
way  ;  hence  they  are  not  necessarily  equal.  A  few  examples  will  render 
the  truth  of  the  lemma  more  clear. 

Ex.  1.   Given  u  =  (x  +  y)m  to  show  that  the  partial  differential 
coefficients  are  equal. 

r>      ^     du  du  ,  Nm_t 

Results,  —  =  m(x  -f  y)      ,  and  —  =  m(x  -f-  y)n  \ 
dx  dy 

Ex.  2.  Given  u  =  log  (x  +  y)  to  show  that  the  partial  differential 

du  1  -  du  1 

coemcients  are  equal.  HesuUs.  —  =  —  •  —  ,  and  3-  =  —  •  —  . 

dx        x  +  y  dy        x  +  y 

Ex.  3.  Given  u  =  tsm~l(x  -j-  y)  to  show  that  the  partial  differential 
coefficients  are  equal. 

du  1  du  1 


Ex.  4.  Given  u=(-)  ,  show  that  the  partial  differential  coefficients 
are  not  equal. 


Ex.  5.  Given  w  =  log  (xy),  show  that  the  partial  differential  coef 

ficients  are  not  equal.  Results,  —  =  -,  and  --  =  -. 

dx       x          dy        y 


76  APPLICATIONS   OF   THE   DIFFERENTIAL   CALCULUS. 

137.  Prob.  —  To  produce  Taylor's  Formula. 

SOLUTION.  —  Let  u  =f(x  -f-  y}  be  the  function  to  be  developed.  It  is  proposed 
to  discover  the  law  of  the  development  when  the  function  can  be  developed  in  the 
form 

u>  =f(x  +  y)  =  A  +  By+Oy*  +  Dy*  +  Ey*  +,  etc.,     (1), 

in  which  A,  B,  C,  etc.,  are  independent  of  y,  and  dependent  upon  x,  the  form  of 
the  function,  and  its  constants. 

Differentiating  with  respect  to  y,  remembering  that  as  A  is  independent  of  y  it 
will  disappear,  and  that  as  the  factors  B,  C,  D,  etc.,  are  likewise  independent  of  y, 
they  are  to  be  regarded  constant,  we  have 

=  B  +  2ty  +  3Dys  +  IE?  +,  etc.     (2). 


Again,  differentiating  with  respect  to  x,  we  have 
du        dA*        dB          dC          dD 

dx  =  5?   +  &*  +  &*  +  & 

Hence  by  (136) 


B  +  20y  +  Wy*  +  4%3  +,  etc.  =         +        y  +  +        ^  +,  etc. 

Now,  by  the  theory  of  development  by  indeterminate  coefficients,  the  coefficients 
of  like  powers  of  y  are  equal,  and  we  have 

B  =  **     2(7=^,    M>  =  £    US-™,   etc. 
dx  dx  dx  dx 

But  as  (1)  is  true  for  all  values  of  y,  we  may  make  y  =  0,  whence  A  =f(x)  =  u', 
letting  u'  represent  the  value  of  the  function  u  when  y  =  0.     Hence  B  =  --, 


_  _  _ 

~~~'         ~  ~~      * 


like  manner  £"  =  T  —  -  —  ?-  —  ,  etc. 
d«4    2-3-4 

Substituting  these  values  of  A,  B,  C,  D,  etc.  in  (1),  we  have 


du 
g- 

which  is  the  formula  sought. 


,  , 

u  =  A,  +  y)  =  u  +  g-     +  & 


138.  Sen.  —  Taylor's  Formula  develops  Ui  =  f(x  -(-  y)  into  a  series  in 
which  the  first  term  is  the  value  of  the  function  when  y  =  0  :  the  second  term 
is  the  first  differential  coefficient  of  the  function  when  y  —  0,  into  y  ;  the 
$i're?  ferw  is  the  second  differential  coefficient  of  the  function  when  y  —  0, 

into  -^—  ;  etc.,  etc. 

1   '  A 

Ex.  1.  Develop  u  =  (jr-ft/)"1  by  Taylor's  Formula,  and  thus  de 
duce  the  Binomial  Formula. 

SOLUTION.  —  Making  y  =  0  we  have  u'  =  a?*.     Differentiating  u  =  xm,  succes 

sively,  we  obtain  —  -  =  ma?"-1,  -  —  =  m(m  —  l)xm-*,  —  -  =  m(m—  l)(m  —  2)x"*-3, 
ox  dx-  dx3 

*  This  is  the  proper  form,  since  A,  B,  C,  etc.,  axe  functions  of  z. 


DEVELOPMENT  OF  FUNCTIONS.  77 

2L  =  m(m  —  l)(m  —  2)(m  —  S)^-4,  etc.     Substituting  these  results  in  Taylor's 

Formula,  we  have 

.   m(m  —  1) 

u  =  (x  -f-  y)»  =  x"1 4-  iw**-»y  H - 

m(m  —  l)(w  —  2)(m  —  3)  .  .  .   .     ,     _.         .  .  _,          , 

— — — : xm~*y4  -f-,  etc.,  which  is  the  Binomial  Formula. 

Ex.  2.  Develop  u  =  log  (x  -f  y). 

SUG'S.— This  being  a  function  of  the  sum  of  two  variables,  we  apply  Taylor's 

du'       1   d*u'  1    d*u        2 

Formula,    u    =  log  x,  --  =  -,  --  =  -  -,  —  =  -,  etc. 


Ex.  3.  Develop  u  =  ax+v. 

loo"2  a         locr3  a 
Result,  u  =  ax(l  +  log  a  y  +  ~|-2/2  +  -~  2/3  +,  etc.). 

Ex.  4.   Develop  u  =  sin  (a?  -j-  y). 


SUG'S.     it'  =  sin  x.  T-  =  cos  a;,  -  —  =  —  sin  x,  -  —  =  —  cos  a,  etc.     Hence 


u  =  sin  (x  +  y)  =  sin  x  -f  cos  a    —  sin  «~  —  cos  ^5-7-3  +  sin  ^.2.3. 


sinx(1  ._ 


1.2.3.4.5-6.7 

since  the  series  in  the  parentheses  are  equal  respectively  to  cosy  and  siny 
(124,  Ex's  3,  and  4). 

Ex.  5.  Develop  u  =  cos  (a?  +  y). 


+}  etc°  = 


3 

cos  a?  cos  i/  —  sin  #  sin  y. 

Ex.  6.   Develop  u  =  sin  (x  —  y),  and  also  u  =  cos  (a?  —  y}. 
Results,    u  —  sin  (x  —  y)  =  sin  x  cos  y  —  cos  x  sin  y,   and  u  = 
cos  (#  —  y)  —  cos  ^  cos  !/ 


Ex.  7.    Develop   u  =  (a:  +  !/)5,    also   w  =  (x  —  y)a,  by  Taylor's 
Formula. 

Ex.  8.  Develop  u  =  (x  —  y)-4. 


78  APPLICATIONS  OF  THE   DIFFERENTIAL   CALCULUS. 

139.  Taylor's  Formula  is  much  used  for  developing  a  function  of 
a  single  variable  after  the  variable  has  taken  an  increment.  When  so 
used  the  increment  may  be  conceived  as  finite  or  infinitesimal,  only 
so  that  it  be  regarded  as  a  variable. 

Ex.  1.  Given  y  =  logx,  to  find  y'  which  represents  the  value  of  y 
after  x  has  taken  the  increment  h. 

SOLUTION,     y  =  log  (a;  -f-  ^)>  which  developed  by  Taylor's  Formula  gives 
y'  =  log  (x  -f  h)  =  log  x  4-  m\^  —  ^  -f  —  —  —  -f-,  etc.  j,  m  being  the  modu 
lus  of  the  system  of  logarithms. 

Sen. — If  h  be  considered  infinitesimal  with  respect  to  x,  so  that  we  have 
h  =  dx,  we  may  drop  all  the  terms  within  the  parenthesis  except  the  first, 

and  write  y'  =  logx  +  --— .     This  is  the  consecutive  state  of  the  function 

CO 

y  =  logo;.     Hence  subtracting  the  latter  from  the  former  we  have  y'  —  y  =• 
dy  —  dlogx  =  -  — .     This  result  is  as  it  should  be,  in  accordance  with  the 

*G 

rule  for  differentiating  a  logarithm. 

Ex.  2.  Given  y  =  3#  —  2x*  —  5,  to  find  y',  which  represents  the 
value  of  the  function  after  x  has  taken  the  increment  h. 

Result,  y'  =  Zx  —  2a?  —  5  +  (3  —  6x*)h  —  12a£  —  12—  =  3^  — 

L  A-O 

2^3  _  5  4.  (3  _  6&)h  _  6tf/i*  —  2fo». 

Sen. — This  result  may  be  easily  verified  by  direct  substitution.  Thus, 
y'  =  3(x  4-  h)  —  2(x  4-  /I)3  —  5.  Expanding,  y'  =  &c  +  3h  —  2a?  —  Gaflh  — 
"OxJts  —  2fr  —  5  =  3x  —  2x*  —  5  +  (3  —  6^)  h  —  6xh*  —  2h\ 


~L4:0.  Prop. — Though  Taylor's  Formula  gives  the  general  form  of 
the  development  of  a  function  of  the  sum  of  two  variables,  there  are 
sometimes  particular  values  of  one  or  the  other  of  the  variables  for  which 
the  development  is  not  true. 

We  will  illustrate  this  proposition  with  a  few  examples. 

Ex.  1.  Develop  u  —  (x  +  y  —  a)"3"  by  Taylor's  Formula,  and  show 
that  the  development  is  false  when  x  =  a. 


.-   —?1  = ' ,  etc.     Hence  substituting  in  Taylor's  Formula, 

,»*"        8<»-a)» 


DEVELOPMENT   OF  FUNCTIONS.  79 

wehavew  =  (z-f?/  —  a)    =(*  —  «)    H  --  7  —  —  ^H  ---  :  --  5—  ,  etc. 

2(x  —  a)       8(x  —  a)2      48(«—  a)1 
Now,  no  absurdity  appears  in  this  series  for  general  values  of  x  ,  but  for  x  =  a 

the  series  becomes  <x,  while  (x  +  y  —  a)  =  y  ,  for  the  same  value.  But  by  hy 
pothesis  x  and  y  are  independent  and  the  development  should  be  true  for  any 
value  of  y  irrespective  of  the  value  assigned  to  x.  Hence  the  conclusion  that  for 

K  __  £?  y%  __  QC  is  contradictory  to  the  hypothesis,  and  false. 

Sen.  —  It  is  evident  that  any  form  of  function  which,  when  developed  by 
this  formula,  gives  a'  factor  of  the  form  (x  =F  a)m  in  the  denominator  of  any 
term  in  the  development,  will  afford  an  instance  similar  to  the  above,  and  the 
development  will  not  be  true  for  x  =  ±  a,  since  for  this  value  (x  =F  a)m  =  0, 
and  the  terms  in  the  denominators  of  which  it  occurs  will  reduce  to  oo. 


Ex.  2.  For  what  value  of  x  is  the  development  of  u  = 
by  Taylor's  Formula,  untrue  ?  -4ns.,  x  =  —  6. 

Ex.  3.  Kequired  the  value  of  the  function  after  x  has  taken  an  in 

crement  h,  when  y  =  b  +  (x  -f-  c)2  -f-  (x  —  a)K  For  what  value  of  x 
does  the  development  fail  ? 

'    Result,  y'  =  b+(x-{-c)*+(x  —  a^+[2(x  +  c)  +  f  (a;  —  afah  + 

i      7i2  3     /i3       , 

[2  +  f  O  _  fl)-T]_  -  K*  _  ar^Ta  +'  etc' 
2/f  =  oo  when  a?  =  a,  and  hence  the  development  fails  for  this  value. 

Sen.  1.  —  If  h  =  dx  the  above  development  is  true  for  all  values  of  x,  for 

then  we  have  y'  —  b  +  (x  +  c)2  +  (x  —  «)*  +  [2(.r  +  c)  -f  f(ar.—  «)*]£,  which 
is  the  same  as  would  be  obtained  by  substituting  x  -f-  li  for  #  in  the  first 
state  of  the  function  and  developing,  and  then  making  h  =  dx,  and  dropping 
the  higher  powers  of  Ji.  For  x  =  a  this  becomes  y'  =  b  -\-  (a  -|-  c)2  -f- 
2  (a  -f-  c}h,  which  is  as  it  should  be,  since  for  x  -f-  h  —  &  +  ^»  y'  =  ^  4~ 
(a  4.  ft  4.  c)2  +  (a  4.  ^  _  a)  ^  =  5  +  az  +  2«7i  +  2ac  +  7^  +  27*c  +  c*  +  ft*  = 
(dropping  higher  powers  of  h)  b  -f  a*  -f  2a^  -f  2<7C  +  27^c•  -f-  c2  =  Z>  + 
(a"  +  2ac  -f  c2)  +  (2aft  +  2cA)  =  5  +  (a  +  c)2  -f  2  (a  +  c)A. 


Sen.  2.  —  It  will  be  observed  that  when  Maclaurin's  Formula  fails  to  give 
the  time  development  of  a  function  it  fails  for  all  values  of  the  variable  ; 
but  when  Taylor's  fails  it  is  only  for  particular  values,  the  general  develop 
ment  being  still  true. 

GENEKAL  SCHOLIUM.  —  There  are  many  other  important  formulae  for  the 
development  of  functions,  but  the  prescribed  limits  of  this  volume  pre 
clude  their  presentation. 


80  APPLICATIONS   OF   THE  DIFFERENTIAL   CALCULUS. 

SECTION  II. 
Evaluation  of  Indeterminate  Expressions, 

141.  The  following  forms  are  called  The  Indeterminate  Forms,  viz., 

-,  ^,  0  X   oo,  oo  —  oo,  0°,   oo«,  1". 

Whenever  an  expression  assumes  any  one  of  these  forms,  the  impor 
tant  question  to  be  determined  is  whether  it  is  really  indeterminate, 
for  it  often  happens  that  the  indetermination  is  only  apparent. 

Of  these  forms,  -  is  the  fundamental  one,  to  which  all  the  others 
can  be  reduced. 

ILL. — Th      -  is  an  indeterminate  form,  is  readily  seen  when  we  observe  that 

the  divisor,  0,  multiplied  by  any  finite  number,  produces  the  dividend,  0. 

We  may  show  that  each  of  the  other  forms  can  be  reduced  to  the  first,  and  hence 
that  they  are  indeterminate  forms.  Thus,  let  a  represent  a  finite  quantity  ;  then 
a  a  • 

-  =  — .  But  -  =  -  X  -  =  r.  That  —  is  an  indeterminate  form  may  also  be 
a  oo  a  0  a  0  oo 

0  0 

seen  directly ;  since  one  infinity  may  be  any  number  of  times  another,  and  the 

symbols  oo  do  not  mean  that  numerator  and  denominator  are  the  same  infinity. 

Again  OX  oo  =  -  X  TT  =  x,  «  being  any  finite  quantity.  Also  oc  —  oo  is  inde 
terminate,  since  the  difference  between  two  infinities  may  be  any  quantity  what 
ever.  Taking  0°  and  passing  to  logarithms,  we  have  0  log  0  =  0( —  oc)  =  —  0  X  co> 

which  has  been  shown  equal  to  -.  Finally,  applying  logarithms  to  00°,  and  I00, the 
former  becomes  0  log  oo  =  0  X  <*>,  and  the  latter  GO  log  1  =  oo  X  0. 

142.  The  apparent  indetermination  often  occurs  from  the  intro 
duction  of  some  hypothesis  which  introduces  a  factor  0,  into  both 
terms  of  the  fraction. 

ILL. — What  is  the  value  of when  x  =  a?    Making  x  =  a  reduces  the 

a  —  x 

expression  to  -  ;  whence  it  would  appear  that  is  indeterminate  for  x  =  a. 

0  a  —  x 

op 3.3 

But  that  such  is  not  the  case  is  evident,  since =  a2  -f-  ax  4-  otfl  which  =  3a2 

a  —  x 

when  x  =  a.  This  apparent  indetermination  arises  from  the  fact  that  the  hypoth 
esis  x  =  a  introduces  a  factor  0  into  numerator  and  denominator.  This  factor 
being  divided  out,  the  true  value  is  seen.  But  it  is  not  always  easy  to  discover 


EVALUATION   OF   INDETERMINATE   EXPRESSIONS.  81 

the  factor  which  becomes  0,  so  as  to  be  able  to  cancel  it ;  hence  the  necessity  of 
some  general  method  of  procedure. 


143.  I*  rob.  —To  evaluate  y  =  -j^for  x  =  a,  when  for  this  value 
of  the  variable  the  function  assumes  the  form  -. 


•     SOLUTION.—  Let  y'  be  the  function  when  x  has  taken  an  increment  h,  so  that 

_  /(a?  +  M  ^     Developing/(«  +  h)  and  q>(x  -f  h)  by  Taylor's  Formula,  and  for 

q>(X  -j-  ft) 
simplicity  using  f(x),f'(x),  ----  <p'(x),  <p"(*\  etc->  for  the  coefficients,  we  have 


_          _ 

*«  +  ">         vtx)  +  ,,-(«,)*  +  V"<*)£i 

But  by  hypothesis,  when  x  =  a,  /(«)  and  <p(x)  each  equals  0.     Hence  dropping 
these  terms  and  dividing  by  h,  we  have 

"a  etc> 


Now  making  h  =  0,  whence  y'  becomes  y,  there  results  y  =  =-—-  =  '—r—y  as  the 
value  of  the  function  for  x  =  a. 

If  however,  =  -,  we  can  drop  the  first  two  terms  of  A,  and  dividing  by 

<p'  (ci)       0 

A2,  making  A  =  0,  and  a?  =  a,  we  have  y  =  . 

Thus  we  can  continue  to  replace  /(«),  and  <p(.r)  by  their  successive  differential 
coefficients  until  a  pair  is  reached  which  do  not  both  reduce  to  0  for  x  =  a.  The 
last  result  will  be  the  true  value  of  the  symbol. 

Ex.  1.  Given  y  = -,  to  evaluate  the  expression  for  x  =  0. 

SUG.    f(x)  =  sin  x,   and  <p(x)  —  x.    f(x)  —coax,  and  tp'(ae)  =  l.     .'.  y  = 
=  i  =  l. 


Ex.  2.  Given  y  =  -    :— k  to  evaluate  for  a:  =  1. 

For  x  =  1,  y  =  1. 


^5 ]_ 

Ex.  3.  Given  y  = T,  to  evaluate  for  a:  =  1. 


For  x  =  1,  y  =  5. 


*  This  subscript  signifies  "  x  being  ==  to  0." 


82  APPLICATIONS   OF  THE  DIFFERENTIAL   CALCULUS. 

Ex.  4.  Show  that  if  x  =  0,  y  =  -        —  =  log  -. 

Ex.  5.  Show  that  if  x  =  a,  y  =  - —  — —  =  oo. 

Suo.    f(x)  =  nenx,  and  <p'(x)  =  s(x— a)'-1.     Hence /'(a)  =  nen<f,  and  <p'(a)  =0. 
.  • .  For  x  =  a,  y  =  —  =  oc. 

•  1 

Ex.  6.  Show  that  if  x  =  0,  y  =  -^ =  7;. 

x*  b 

StiG. — The  first  and  second  differential  coefficients  of  both  numerator  and  de 
nominator  reduce  to  0  ;  but  f"(x)  =  cos*,  and  <p'"(x)  =  6.    Hence  for  x  =  0, 
cosx       1 
6          6 

Ex.  7.  Evaluate  y  =  — — ~ —  for  x  =  0.  yx=  0  =  2. 

#  —  sin  # 

Ex.  8.  Evaluate  y  =  — ^^—  for  a?  =  1.  yXBSl  ==  0. 


*1   r 

Ex.  9.  Evaluate  y  =  -  ~  --  for  x  =  0.  ^_o  =  J. 


_ 

Ex.  10.  Evaluate  y  =  —       —  for  ^  =  a. 


SUQ.  —  For  x  =  a,  the  first  and  all  succeeding  differential  coefficients  of  both 
numerator  and  denominator  reduce  to  <x>.  Hence  we  see  that  for  x  =  a  the  devel 
opment  of  these  functions  by  Taylor's  Formula  is  not  true.  Moreover,  if  it  were 

true,  we  should  but  exchange  the  symbol  -  for  —  .     In  this  case,  however,  it  is 

0          oo 

easy  to  see  the  factor  which  gives  the  expression  the  indeterminate  form.     It  is 
(a  —  a)*.     Cancelling  it,  y  =  (a  +  »)L*  =  (2«A 

v  x  —  v  a  -f  V  x  —  a  ., 
Ex.  11.  Evaluate  y  =  -  ==  --  for  x  =  a. 

V  x*  —  a2 


Suo.— This  example  is  like  the  preceding.     But  dividing  by  \/x  —  a,  we  have 

PL  +  1 

N/2v/a  1 


y  = 


v/2a  v/2a 


EVALUATION   OF  INDETERMINATE  EXPRESSIONS.  83 

144,  Prob*  —  To  evaluate  j  =  —  ~-/or  x  =  a,  when  for  this  value 
of  the  variable  the  function  assumes  the  form  —  . 


SOLUTION,    y  =  ——  =  ^^-  =  -,  when  /(a;)  and  <p(x)  are  each  oo.    Now  apply- 

<p(X) 

A*) 

1 

ing  to  y  =  2-f»  the  method  of  the  preceding  problem,  we  have 


/(») 

f(a») 

Dividing  the  first  and  last  members  by  -  -,  we  have 


whence  « 

/(x)    A  '  - 


Therefore  the  process  in  this  case  is  the  same  as  in  the  preceding. 

0 

Ex.  1.  Evaluate  y  =  — ~-  for  x  =  oo. 

1 

Suo.    yx=a.  =  JL  =  _1  =  J_  =  a 

nxn~l        nxn        n  oon 

Ex.  2.  Evaluate  y  =  2f-^  for  #  =  0. 

cot  # 

1  l 

x  x  sin2x       0      m. 

Suo.    3^=0  = =  — 7—  = =  rt-     Therefore  differentiating  again, 

cosec^a;  1  x          0 

sin2x 
2  sin  x  cos  ar        0 

y*=<>  =  — j =  i  =  o. 

i  loo1  '/* 

Ex.  3.  Evaluate  y  = i  -  for  a;  =  oo.  ys=ao  =  0. 


Ex.  4.  Evaluate  t/  =  —  for  x  =  oo. 

£ 

Suo. — As  the  successive  differential  coefficients  continue  to  be  oofor  »=  oo  until 

n(n  — l)(n  — 2) 3-2-1 


we  reach  the  nth,  we  differentiate  n  times  and  obtain  y  = 

=  0,  when  x  =  oo. 


e* 
n(n  —  l)(n  —  2) 3  •  2  •  1 

00 


84  APPLICATIONS   OF  THE   DIFFERENTIAL   CALCULUS. 


Ex.  5.  Evaluate  y  =  —  —  for  x  =  0.  yx^  =  —  . 

.   TtX  O 


Ex.  6.  Evaluate  y  =  loftan(2a)  for  x  =  0.  y.  .  =  I. 

log  tan  x 


14:5*  I*TOb. — To  evaluate  j  —  f(x)  x  cp(x]  for  x=  a,  when  for  this 
value  of  the  variable  the  function  assum.es  the  form  0  X  oo. 

SOLUTION. — Since  the  reciprocal  of  that  function  of  x  which  becomes  oo,  is  0, 
we  may  write  y  =f(x)  X  <p(%)  =•  :A —  =  r^  for  x  =  a,  f(x)  being  0,  and  <p(x) 


being  oo.     Therefore,  putting  the  expression  in  the  form  y  =  ~-T— • ,  it  may  be 
treated  as  the  first  case  (JE43). 

Ex.  1.  Evaluate  y  =  2*  sin  —  for  x  =  oo. 

.    a 
sin— 

SUG.     Since  for  x  =  oo,  2*  =  oo,  and  sin  —  =  0,  we  write  y  =  2*  sin  —  =  ~^_^-- 
Whence  replacing  the  numerator  and  denominator  by  their  differential  coefficients, 

—  «2~*  log  2  cos  |^  ^ 

we  have  y  = ; • =  a  cos  —  =  a,  when  x  =  oo. 

—  2~x  log  2  2X 

Ex.  2.  Evaluate  y  =  (1  —  x)  tan  —  for  x  =  1. 

SUG. — Since  tan  -—  =  00  when  x  =  1,  we  write  t/  =  — —  =          '    (differen- 


_  -i  o  2 

tiating)  =  -  =  -  =  -,  when  x  =  1. 

?T  7T.T  Tt  It 

—  —  cosec2  —        Tt  cosec2  — 


Ex.  3.  Evaluate  y  =  ez  sin  x  for  #  =  0. 


SUG.     y  =     sin«=—     ==       l     ==  z  V  cos  a:  ==  ^  e*  =  0  X  oc,  when  x  =  0. 

«;    i«" 
i 

Were  we  to  repeat  the  process  upon  x*ex  we  should  find  that  its  form  would  re- 


EVALUATION   OF   INDETERMINATE   EXPRESSIONS.  8 

main  the  same.     But  put  -  =  z,  whence  &e  =  ^,  and  differentiating  twice,  w 


1 

x  1  1 


e        e  -  -       e 

have  -  =  —  =  oo,  when  x  =  0.    .  •  .  y  =  ex  sinx  =  x*ex  =  -  =  oo,  when  x  =  0. 

Ex.  4.  Evaluate  y  =  xm  log"  x  for  x  =  0. 

SUG.    y  =  —  f—  =  —  ,  when  a;  =  0.     Now  differentiating,  yxe=0  =  -  ?  = 

oc  m 

OJ"1  ~x»<+l 

n(n  —  l)loe"-*o;  .  _ 

(differentiating  again)  =  -  x  =  n("~  1}  lo*~x.     Aftei 

?n'-'  * 


»  differentiations,  we  hare  „,_.  =  Nn  -  l)(n  -  2) 
n(n  —  l)(n  —  2)  .....  3-2. 


[It  is  sometimes  expedient  to  put  the  function  in  the  form  —  rather  than  -. 

oo  0 

Experiment  must  decide  which  is  preferable  in  any  given  case.  ] 


140.  Prol).  —  To  evaluate  y  =  f(x)  —  <p(x)  for  x  =  a,  when  for 

this  value  of  the  variable  the  function  assumes  the  form  oo  —  oo. 

SOLUTION.  —  Since  /(x)  =  oo,  and  <p(x)  =  oo,  we  may  write  y  =  -  ----  = 

/(*)         <P$) 

—  -  —  -  -  =  -.     Having  put  the  function  in  the  latter  form  it  may  be  treated 


f(x) 
as  in  the  first  case  (143). 

2  1 

Ex.  1.  Evaluate  y  =  --  -  ---  =•  for  x  =  1. 

or-'  —  1       x  —  1 

SOLUTION.  —  In  this  case/(x)  =  -  -,  and  tp(x)  —  --  .     Hence  -:  —  =  —  HT_, 

o 


= .      We  may   therefore    write    v  = — — 

<p(x)  1  y         &  —  1 


(differentiating)  = = . 


[In  this    case  the   factor  1  —  x  can  be  divided  out  without    differentiating. 

...  2  1  2-a;-l        1—x  1 

Moreover  -—  -  _  =  — _-  =  -—  ==  _  ._.] 


85  APPLICATIONS   OF  THE  DIFFERENTIAL   CALCULUS. 

Ex.  2.  Evaluate  y  =  —^  —  -—  for  x  =  1.  y,=1  =  i 

x  —  1        log  x  2 

Ex.  3.  Evaluate  y  =  sec  a;  —  tan  x  i or  x  =  -. 

A 

Suo. — This  may  be  treated  exactly  as  the  last;  but  the  following  is  more  ele- 

1  sin  x         1  —  sin  x         0  TT 

gant.     y  =  sec  x  —  tan  a;  = =  =  -,  when  x  =  -. 

cosx         cosx  cosiC  0  2 

Whence,  differentiating,  y     ^  = : =  0.     Therefore  when  a;  =  ^',  sec  x  and 

z= Sin  X  A 

3 

tan  x  are  equal,  a  fact  not  difficult  to  observe  from  a  figure. 

1  x 

Ex.  4.  Evaluate  y  =  = — —  for  a;  =  1.  y,-i  =  —  1. 

log  x       log  x 


'.  JProb.—To  evaluate  y  ==  {f(x)}^(x)/or  x  =  a,,  when  for  this 
value  of  the  variable  the  function  assumes  either  of  the  forms  0°,  00°, 
or  1". 

SOLUTION. — Passing  to  logarithms  we  have  log  y  =  ((px~)  log/(x).  When 
/(x)  =  0  and  <p(x)  —  0,  logy  =  <p(x)  logf(x)I=a  =  0  X  (—  oo)  ;  when/(x)  =  oo  and 
<p(x~)  =  0,  logy  =  <p(x)  log/(x)I==a  =  0  X  oo  ;  when  f(x}  —  1  and  <p(x)  =  oo, 
log  y  =  <p(x)  log/(x)x=a  =  oo  X  0.  Hence  all  these  cases  fall  under  (145}. 

Ex.  1.  Evaluate  y  =  af  for  a:  =  0. 

SOLUTION,     log  y  =  x  log  .r  =  — y-  =  -  when  x  =  0.     Whence,  differentiating, 

x 

log  yx=0  = j =  —  x  =  0.     Finally,  as  log  y  =  0,  y  =  1. 

"aT^ 


Ex.  2.  Evaluate  y  =  af101,  and  y  =  (sina?)'1"  for  a;  =  0. 

Sua.  —Since  for  x  =  0,  sin  x  =  x,  these  are  each  =  x*.     .• .  yx=s0  =  (sin  x)8ln  x  = 
a?  —  1,  by  Ex.  1. 

These  may  also  be  solved  directly.     Thus  y  =  x8in  x,  gives  log  y  =  sin  x  log  x  = 

°°x  .  — ,  when  x  =  0.     Hence  by  (14&),  and  differentiating  twice,  we  have 

cosec  x  co 

1 

log  x  x  sin2  x  2  sin  x  cos  a;  2  sin  x  cos  x 

gyz=o          i  cos  a?  iccosa:  cosx  —  xsin#  cosx 

sinx  sin-  x 

2  sin  x  =  0.     .  • .  y  =  1. 


EVALUATION   OF   INDETERMINATE   EXPRESSIONS.  87 

Also  y  =  (sin xisi" x  gives  logy  =  sin x  log  sin  x  =  —          - —  (differentiating) 

COStJC  iC#=  o 

cotx  1 

-  =  0.     .  • .  y  —  1. 


cosec  #  cot  x  co 

Ex.  3.  Evaluate  y=  (cotx)»iux  for  a;  =  0. 

SUG. — Put  this  in  the  form  — .     Thus  log  v  =  sin  x  log  cotx  =  — (dif- 

oo  cosec  xz=o 

cosec2  x 


cot.c  cosec  x         smx         0 

ferentiating)  =  = =  — >—  =  -  =  0.     .  • .  y  =  1. 

—  cosec  x  cot  x         cot-  x         cos-ac        1 


Ex.  4.  Evaluate  y  =  (1  -f  nx)1  for  x  =  0. 

Sua.     log  y  =  — - — i =  Differentiating,  log  3^=0  =  -   =  n. 

^  -c=o  1 

.  • .  y  =  e". 

Ex.  5.  Evaluate  y  =  {cos(aar)}coflec2lcr)  for  a?  =  0. 

SUG.     y  =   {cos(#x)}cosec2(e:c)  =  1°°,    when   x  =  0.      Passing  to  logarithms 

log  y  =  cosec2  (ex)  log  cos  (ax)  =  —. -1 — -  =  -z,   when  x  =  0.     Differentiat- 

sin*  (ex)  0 

—  a  tan  (ax)  a  tan  (ax)  _  —  a2  sec2  (ax)  a* 

og  yx=0  — 


APPLICATIONS  OF  THE  DIFFERENTIAL  CALCULUS. 

SUCTION    III. 
Maxima  and  Minima  of  Functions  of  One  Variable, 

148.  DEF.  —  A  Maximum  value  of  a  function  of  a  single  vari 
able  is  a  value  which  is  greater  than  the  immediately  preceding  and 
the  immediately  succeeding  values  ;  i.  e.,  the  value  when  the  variable 
takes  an  infinitesimal  decrement,  and  the  value  when  the  variable 
takes  an  infinitesimal  increment. 


'S.  —  Let  y  =  sin  x.  When  x  =  —  ,  y  is  a  maximum,  since  it  is  greater  than 
the  immediately  preceding  and  the  immediately  succeeding  values.  If  x  takes  an 
increment  h,  making  y'  =  sin(:  —  f-  h\  or  a  decrement,  —  h,  so  that  y"  = 

sin^-  —  h\  y  is  evidently  greater  than  y'  and  y",  as  at  90°  the  sine  is  greater 

than  it  is  at  a  little  more  or  a  little  less  than  90°. 

Again,  constructing  the  equation  y*  =  6x2  —  #3,  we 
find  the  right  hand  branch  to  be  as  given  in  the  figure. 
Here  y  =/(#),  and  y  is  a  maximum  when  x  =  A  D  = 
4,  since  for  x  infinitesimally  less  or  greater  than  4,  y  is 
less  than  for  x  =  4.  The  maximum  value  of  y  is, 
therefore,  y  =  tyii  .  4*  —  43  =  34  nearly.  FIG 

Once  more,  let  y  =  8x  —  ac8.      If  x  =  1,  y  =  7  ;   if 

x  =  2,  ?/  =  12  ;  if  x=3,  y  =  l5  ;  if  x  =  4,  y  =  16  (a  maximum)  ;  if  cc  =  5,  y=15  ; 
if  x  =  6,  y  =  12  ;  and  if  x  =  7,  y  =  7.  Hence  it  appears  that  as  x  increases  y  in 
creases  till  it  has  attained  a  certain  value,  when  although  x  is  made  to  continue 
its  increase,  y  begins  to  diminish.  The  point  at  which  the  function  ceases  to  in 
crease  and  begins  to  decrease  is  its  maximum.  In  this  case  it  will  be  found  that 
however  little  x  varies  from  4,  either  way,  y  becomes  less  than  16.  Thus  if  x  = 
3.9,  y  =  15.99  ;  and  if  x  =  4.1,  y  =  15.99. 

14:0.  DEF.  —  A  ]}linimu/ni  value  of  a  function  of  a  single  vari 
able  is  a  value  which  is  less  than  the  immediately  preceding  and  the 
immediately  succeeding  values;  i.e.,  the  value  when  the  variable 
takes  an  infinitesimal  decrement,  and  the  value  when  the  variable 
takes  an  infinitesimal  increment. 

ILL'S.  —  Let  7/  =  cosecx.  As  x  approaches  —  y  diminishes  and  approaches  1, 
reaching  1  at  a;  =  -  .  When  x  passes  —  ,  y  begins  to  increase,  so  that  y  =  1,  is  a 

a  2 

minimum  value  of  the  function  y  =  cosec  x. 

Again,  y  —  x2  —  6x  -f-  10,  has  a  minimum  value  for  x  =  3,  at  which  value 


MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  ONE  VARIABLE.    89 


y  -=  1.  By  substituting  values  of  x  a  little  greater  than  3, 
as  3.01,  and  a  little  less  as  2.09,  y  will  be  found  to  be  greater 
than  1  in  both  cases.  The  locus  of  the  function  is  given  in 
Fcg.  20,  where  P  D  represents  the  minimum  value  of  y. 


M\|Y 


/N 


COR. — The  same  function  may  have  several 
maxima  or  several  minima  values,  and  these  may  be 
equal  or  unequal.  Moreover,  a  maximum  value  may 
be  equal  to  or  even  less  than  a  minimum  value  of  the 
same  function.  FIG.  20. 

ILL'S.— The  function  y  ==  x4  —  8z'  -f-  22x2  —  24.r  -f  12,  has 
minima  values  for  a;  =  1,  and  »  =  3,  which  values  are  both 
y  =  3  ;  or  two  equal  minima  values,  as  illustrated  by  the  ordi- 
nates  at  P  and  P"  in  the  figure.  For  x  =  2  y  =  4,  a  maximum 
value,  as  illustrated  by  P'D'. 

Again,  let  y  ==  f(x)  be  the  equation  of  M  N   referred  to  A  X 
and  AY  Fig.  22.     Then  PD,  P"D",  and  P^Dlvare  maxima 
values   of  y ;    and 'P'D',   and    P'"D'"   are  minima  values. 
But  the  several  maxima  values  are  unequal  and  the  minimum     A  D  D'  D" 
P'  D'  is  greater  than  th'e  maximum  P'vDlv.  FIG.  21. 


Sen.  —  It  will  be  observed 
that  the  terms  maximum  and  mini 
mum,  as  here  used,  do  not  mean  the 
greatest  possible  and  least  possible. 
Thus,  if  we  ask  for  the  maximum 
value  of  y  in  y  =  x3  —  3ax*  —  5,  we 
do  not  inquire,  what  is  the  greatest 
possible  value  which  y  can  have  ? 
but  simply,  whether  if  x  vary  con 
tinuously  through  all  possible  values, 
there  is  any  point  at  which  y  will  at 

tain  a  greater  value  than  it  had  immediately  preceding  that  point,  and  tlinn 
it  will  have  immediately  after  passing  that  point  ;  and,  if  there  be  such  a 
value  of  y,  what  it  is. 


FIG.  22. 


.  Prop. — In  an  explicit  function  of  a  single  variable,  y  =  f(x), 

dy 

the  first  differential  coefficient,  y-  changes  sign  from  -f  to  — ,for  contin 
uously  increasing  values  of  the  variable,  where  the  function  is  at  a  maxi 
mum,  and  from  —  to  +  where  the  function  is  at  a  minimum.  Hence  for 
such  values  the  first  differential  coefficient  =  0  or  oo. 

DEM. — Let  y  =f\x}  be  the  function.     Fir.ft,  For  x  =  x',  suppose  y  becomes  y'.  a 
maximum.     Then  y'  =  f(x')  is  at  a  maximum.     Now  the  immediately  preceding 


90  APPLICATIONS   OF   THE   DIFFERENTIAL   CALCULUS. 

state  of  the  function  is  f(x'  —  dx'\  and  we  have  —  =  /(^  —  dx')  —fix'} 

dx  (x'—dx)  —  x 

hypothesis  f(x  —dx')  —f(x')  is  —*,  and  as  (x  —  dx')  —  x'  is  evidently  —  ,  we  have 

^7  +.     Again,  the  immediately  succeeding  state  to  y'  =f(x)  is/(x'-f-dx')  —  /(x'); 


hence  we  have  =-.  By  hypothesis  f(x  +  dx')  -f(x)  is  -*, 
and  as  (x'  -f-  dx')  —  x'  is  evidently  -f-,  we  have  j-,  —  .  Therefore  where  y'  = 
f(x')  is  a  maximum  -=-,  changes  sign  from  -f-  to  —  . 

Second.  If  y'  =f(x')  is  at  a  minimum  ^  ='f(a;>  ~  dx'l  ~  W>  is  _,  since  by 

dx          (x'—dx'  )—x' 

hypothesis  f(x  —  dx')  —f(x')  is  -f-,  and  (x  —  dx')  —  x1  is  evidently  —  .  Again 
df7  =  ('~-d'~-'}  is  +'  as  by  Hypothesis  /(«'  -f  dx')  —/(a?')  is  +,  and 


(x'  -f-  dx')  —  #'  is  evidently  -f  . 
Finally,  since  when  a  varying  function  changes  sign  it  passes  through  0  or  oo, 

we  have  -^-  =  0  or  oo  for  maxima  and  minima  values  of  the  function.     Q.  E.  D. 

GEOMETRICAL  ILLUSTRATION.  —  TS  being 
tangent  to  the  curve  M  N  at  P,  P'  being  a 
consecutive  point  so  that  P  E  represents  dx, 
and  P'E  dyt  we  observe  that  the  angle 
P'PE  =  a,  the  angle  which  the  line  makes  ^^  / 

with  the  axis  of  abscissas.     Hence  tan  a.  = 


DD' 
tan  P'PE  =  ^£  =  jg;  i  e.  the  first  dif-     'R 

ferential  coefficient  of  the  ordinate  regarded  FIG.  23. 

as  a  function  of  the  abscissa,  represents  the  tangent  of  the  angle  which  a  tangent 

to  a  plane  curve  makes  with  the  axis  of  abscissas. 

Now,   observing   Fig.  22  we  see  that  as  x  is  increasing,  and  y  approaching  a 
maximum  value  as  PD,  the  tangent  to  the  curve  makes  an  acute  angle  ;  hence 

approaching  P  from  the  left  --  is  -f .     At  P  the  tangent  becomes  parallel  to  the 
axis  of  x ;  tan  a  =  -^-  =  0.     Immediately  upon  passing  P,  a  becomes  obtuse, 

and  consequently  tan  a  =  ~  is  — . 

So  also  in  approaching  a  minimum  value  as  P'  D'  from  the  left  it  appears  that 
a  is  obtuse^  and  hence  --  —  ;  at  this  point,   P',  a  —  0,  and  -  -  =  0  ;  and  after 

passing  P',  a  becomes  acute  and  --  -{-. 


*  The  hypothesis  is  that  y'  =J[x')  is  a  maximum,  i.  e.  is  greater  than  either  the  immediately 
preceding  arid  the  immediately  succeeding  states  of  the  function.  But  f(xf  —  dx')  is  the  imme 
diately  preceding  state,  andy|z'  4-  dx)  is  the  immediately  succeeding  state.  Hence f(x'  —  dx')  <^ 


D 


MAXIMA   AND   MINIMA   OF   FUNCTIONS   OF   ONE   VARIABLE.  91 

To  illustrate  the  case  in  which  --  changes  si<m 

dx 

by  passing  through  oo,  consider  y  =f\x)  as  the 
equation  of  MN,  fig.  24.  PD  is  evidently  a 
maximum  ordiuate.  But  in  approaching  PD 

from  the    left,    a  is  an  acute  angle,  and  --,   -(-. 

At  P,  a  =  9(P,  ana  (V=  ^     After  passing  PD,  FlG-  24- 

a  is  obtuse  and  --,  — .     A  similar  illustration  may  be  given  of  the  case  in  which 

~  passes  through  oo  at  a  minimum. 

Sen. — The  student  needs  to  guard  against  the  error  of  supposing  that  all 
values  of  the  variable  which  render  the  first  differential  coefficient  0  or  oo, 
necessarily  render  the  function  a  maximum  or  minimum.  These  values  of 
the  variable  correspond  to  the  maxima  and  minima  values  of  the  function  if 
it  has  any  maxima  or  minima  values,  since  if  the  first  differential  coefficient 
changes  sign,  it  must  pass  through  0  or  oo  ;  but  a  quantity  may  pass  through 
0  or  oo  without  changing  sign,  so  that  the  values  of  the  variable  which 
render  the  first  differential  coefficient  0  or  oo  are  simply  critical  values,  i.  e. 
values  to  be  examined. 

153.   Prop* — In  an  explicit  function  of  a  single  variable,  y=f(x), 

d2y 

the  second  differential  coefficient,  — -)}  if  not  0,  is  —  where  the  function 

is  at  a  maximum,  and  +  where  it  is  at  a  minimum. 

DEM. — Let  y  =f(x)  be  the  function.  We  have  seen  that  when  the  function 
passes  through  a  maximum  --  changes  sign  from  -f-  to  —  for  continuously  in 
creasing  values  of  x,  L  e.  —  is  decreasing  /  and  when  the  function  passes  through 

a  minimum  --  changes  sign  from  —  to  4-,  i.  e.  --  is  increasing.     Now  — -  = 
dx  dx  dx2 


dx         df(x)        f(x  +  dx}  —f  (a-.)      . .  ,   . 

-y-  =  -^ =  -j ,  which  is  —  when  the  numerator  is  — ,  and 

dx  dx  dx 

-j-  when  the  numerator  is  -f ,  since  dx  is  -{-  by  hypothesis.  But  at  a  maximum  -- 
is  decreasing  for  increasing  values  of  x,  and  f(x  -\-  dx)  — f(x)  is  —  ;  and  at  a 
minimum  --  is  increasing  for  increasing  values  of  x,  and/' (x  -f-  dx)  —  f(x}  is  +  . 

Therefore  —  is  —  at  a  maximum  value  of  the  function  and  -{-  at  a  minimum, 
unless  it  is  0,  a  case  which  is  not  yet  provided  for. 

Sen.  1. — The  ordinary  method  of  examining  an  explicit  function 


92  APPLICATIONS   OF  THE   DIFFERENTIAL   CALCULUS. 

of  a  single  variable  for  maxima  and  minima  values  is  to  form  the  first  differ 
ential  coefficient,  put  it  equal  to  0,  and  solve  the  resulting  equation.  Some  or 
ah1  of  the  values  of  the  variable  thus  found  may  correspond  to  maxima  and 
minima  values  of  the  function.  They  are  then  to  be  examined  separately. 
To  do  this,  form  the  second  differential  coefficient  of  the  function  and  sub 
stituting  in  it  the  value  of  the  variable  to  be  examined,  if  it  gives  a  _  re 
sult,  this  value  of  the  variable  corresponds  to  a  maximum  value  of  the 
function  ;  but  if  it  gives  a  -f  result,  it  corresponds  to  a  minimum  value  of 
the  function.  Thus  all  the  values  of  the  variable  arising  from  equating  the 
first  differential  coefficient  with  0,  are  to  be  examined.  If,  however,  any  one 
of  these  critical  values  renders  the  second  differential  coefficient  0,  it  is  best 
to  examine  the  first  differential  coefficient  for  this  value  and  see  if  it  actu 
ally  does  change  sign  in  passing  from  a  value  of  the  variable  infinitesimal!}1 
less  to  a  value  infinitesimally  greater  than  that  being  examined. 

155.  Sen.  2.—  The  following  axiomatic  principles  often  facilitate  the  ex 
amination  of  a  function  for  maxima  and  minima  values  : 

1st.  Whatever  value  of  x  renders  u  =  f(x)  a  maximum  or  minimum,  ren 

ders  u'  =  af(x)  or  u"  =  -±-i  a  maximum  or  minimum.     Hence  constant  fac- 
a 

tors  or  divisors  may  be  dropped  from  the  function. 

2nd.  Whatever  value  of  x  renders  u  =  f(x)  positive  and  a  maximum  or 
minimum,  renders  u'  =  [/(«)]"  a  maximum  or  minimum,  n  being  a  positive 
integer  ;  but  if  u  =f(x)  is  rendered  negative  for  the  particular  value  of  :c, 
u<  —  [/(^)]2n  is  a  minimum  when  u  =f(x)  is  a  maximum,  and  a  maximum 
when  u  =f(x)  is  a  minimum.  Hence  the  function  may  be  involved  to  any 
power. 

3rd.  Whatever  value  of  x  renders  u  =  log  [/(#)]  a  maximum  or  minimum 
renders  u  —  f(x]  a  maximum  or  minimum.  Hence  to  examine  the  log 
arithm  of  a  function  we  have  to  examine  simply  the  function  itself  dropping 
the  symbol  log. 


Ex.  1.  What  values  of  x  render  y  =     ktfx*  —  'lax*  a  maximum  or 
minimum  ;  and  what  are  the  maxima  and  minima  values  of  y  ? 


SOLUTION.  * — Whatever  value  of  x  renders  y  =  \/4a-x-  —  '2ax*  a  maximum  or 
minimum  renders  y2  or  y'  =  4a%*  —  2«a;:i  a  maximum  or  minimum  (155}.  And 
fora  similar  reason  we  may  drop  the  constant  factor  2a,  and  examine y"  =  2ax2  —  x*, 
since  any  value  of  x  which  renders  this  a  maximum  or  minimum  °will  also  render 

the  original  function  a  maximum  or  minimum.  Differentiating  we  have  ~—  — 
4ax  —  3x2.  Now  whatever  value  of  x  renders  the  function  a  maximum  or  mini 
mum  renders  4aa  —  3x2  =  0.  From  this  x  =  0,  x  =  -- .  If,  therefore,  there  are 

o 

any  maxima  or  minima  values  of  the  function,  they  are  those  which  correspond  to 

*  This  solution  may  seem  needlessly  prolix,  but  the  author  finds  that  comparatively  few  stu 
dents  really  follow  the  argument  through  unless  required  to  give  it  thus  in  detail. 


MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  ONE  VARIABLE.     93 

one  or  the  other  or  both  of  these  values  of  x.     Differentiating  again,  —  ^  =  4.a  —  6x. 
For  x  =  0,  —  =  4a  ;  hence  x  =  0  corresponds  to  a  minimum  value  of  the  func 

tion.     For  x  =  -3-,   —  =  4a  —  8a  =  —  4a  ;  hence  x  =  ~  corresponds  to  a  max- 
imuni  value  of  the  function. 

Substituting  these  values  of  x  we  find  y  =  v/4a*sca  —  2«.-r:i  =  0,   a  minimum 

value  ;  and  y  =  v/4a***  -  Hoa*  =      fef!  _  1*^  =   **_,  a  maximum. 

N    9  27          3^3 


Ex.  2.  What  values  of  x  render  y  =  &  —  9^2  _j_  24ar  —  16  a  maxi 
mum  or  a  minimum,  and  what  are  the  maxima  values  of  y  ? 

Results,  x  =  2  corresponds  to  a  maximum,  and  #  =  4  to  a  mini 
mum.     The  maximum  value  is  y  =  4,  and  the  minimum  y  =  0. 

Ex.  3.  Examine  y  =  &  —  3x*  —  24j?  -f  85  for  maxima  and  minima. 
Results,  For  x  =  4,  y  —  5,  a  minimum  ; 

For  #  ='  —  2,  y  =  113,  a  maximum. 
Ex.  4.  Examine  y  =  5  (a?  —  #2)  for  maxima  and  minima. 

SUG.—  Drop  the  5.     x  =  i,  gives  y  =  f,  a  maximum. 

Ex.  5.  Examine  ?/  =  (2ax  —  x*}~*  for  maxima  and  minima. 

SUG.—  Use  y'  =  2az  —  jc2.     x  =  a,  gives  y  =  a,  a  maximum,  and  —  a,  a  min 


imum. 


Ex.  6.  Examine  y  =  x4  —  8^3  +  22#'  —  24#  -f  12  for  maxima  and 
minima. 


dl  =  4ic3  ~  24r2  +  44a;  —  24  =  °>  or  ^  —  6x2  +  Ha;  —  6  =  0.  To 
find  the  roots  of  this  equation,  observe  that  the  factors  of  the  absolute  term  with 
its  sign  changed  are  1,  2,  and  3  (COMPLETE  SCHOOL  ALGEBKA,  111}.  By  trial 
these  are  found  to  be  the  values  of  x,  x  =  1  gives  y=.  3,  a  minimum  ;  x  =  2  gives 
y  =  4,  a  maximum  ;  x  =  3  gives  y  =  3,  a  minimum  (see  ILL.  Fig.  21). 

Ex.  7.  Examine  y  =  x*  —  5&  +  5.r3  +  1  for  maxima  and  minima. 

Results,  The  critical  values  of  x  are  0,  0,  1,  3.  For  x  =  1,  y  —  2, 
a  maximum  ;  for  x  =  3,  ^/  =  —  26,  a  minimum,  x  —  0  does 
not  correspond  to  either  a  maximum  or  minimum  value  of  y. 

SUG.—  That  x  =  0  does  not  correspond  to  either  a  maximum  or  a  minimum  is 
determined  as  follows  : 


Having  J  =  5x4  ~  20x3  +  15x2'  substitute  0  —  h  and  0  -f  h  for  x,  and  evaluate 
the  expression  for  ^infinitesimal,  thus  determining  whether  -^  changes  sign  or  not, 


94  APPLICATIONS   OF   THE   DIFFERENTIAL   CALCULUS. 

in  passing  through  x  =  0.     Thus  —•  =  5(0  —  h)4  —  20(0  —  h)3  4.  15(0  —  h)2  = 

5h*  4-  20fc3  4-  15/t2  =  15h2,  when  h  is  infinitesimal.    Again  ~    =  5/i4  —  20fc3 

dy 

4-   15/i,2  =  15/i*,  when  h  is   infinitesimal.      Therefore,  as   --  has  like  signs  on 

dx 

both  sides  of  x  =  0,  and  consecutive  with  it,  it  does  not  change  sign  in  passing 
through  x  =  0.  Hence  x  =  0  does  not  correspond  to  either  a  maximum  or  a  min 
imum. 

Ex.  8.  Examine  y  =  b  4-  \x  —  a)3  for  maxima  and  minima. 

SUG'S.      --  =  3(x  —  a)2  =  0,  gives  x  =  a.     Hence  if  there  is  any  maximum  or 

minimum  it  must  be  y  =  6,  as  no  other  value  of  x  than  x  =  a  will  render  —  =  0. 

dx 

Again,  since  this  value  renders  —  =  0,  we  examine  it  by  ascertaining  whether  -^ 
changes  sign  at  x  =  a.  -j-  =  3 (a  —  7i  —  a)2  =  3/i»  is  the  value  of  -^  next  preced 
ing  x  =  a  ;  and  -j-  =  3(a  -f-  h  —  a)2  =  3/i2  is  the  next  succeeding  value.  There 
fore,  as  --  does  not  change  sign  at  x  =  a,  the  function  has  no  maximum  nor  mini 
mum  value. 

Ex.  9.  Examine  y  =  a(x  —  6)4  4-  c  for  maxima  and  minima  values. 

SUG'S.     -~  =  4(aj  —  6)3  =  0.     .  • .  x  =  6.     —  =  4(6  —  h  —  6)3  =  —  4/i3,  and 

-f-  =  4(6  -f-  h  —  6)3  =  4/i3,  are  the  values  of  — -  immediately  preceding  and  suc- 
clx  ax 

ceeding  x  =  6  ;  hence,  as  — —  changes  sign  from  —  to  -f-  at  this  point,  x  =  6  cor 
responds  to  a  minimum.  .  • .  y  =  4a(6  —  6)4  -\-  c  =  c  is  B,  minimum. 

Ex.  10.  Examine  y  =  (x  —  l)4(x  4-  2)3  for  maxima  and  minima. 

SUG'S.     -^  =  4(o?  —  1)3(#  -f  2)3  -f-  tyx  —  l)4(x  4-  2)2  =  {(x  —  l)3(x  4-  2)-'} 

{4^4-2) -f  3(x —  1)}  =  (x  —  l)3(a;4-2)-->(7a;4-5)  =  0.  .-.  #  —  1  =  0,  #4-2  =  0, 
7.c  4-5  =  0,  give  x  =  l,  x  =  —  2,x  =  —  ^-as  the  critical  values  of  x. 

fy|  =  3(x  —  l)2(x4-2)2(7x4-5)4-2(x  — I)3(x4-2)(7x-f5)4-7(x  — l)3(x4-2)2  =  0 

(123)    92 
for  x  =  1,  and  x  =  —  2,  but  is  — for  x  =  —  f .     The  latter  value,  there- 

4 

fore,  corresponds  to  a  maximum,  and  gives  y  =  ( — $ —  1)4( — f  +  2):'  =  ,  a 

maximum. 

To  ascertain  whether  x  =  1  corresponds  to  a  maximum  or  minimum,  notice 


MAXIMA  AND   MINIMA   OF   FUNCTIONS   OF  ONE   VARIABLE.          95 
that       =  (1  —  h  —  1)3(1  —  h  +  2)2(7  —  7/i  -f-  5)  =  —  fr»(3  —  /*)2(12  —  7h)  is  —  , 


and        =  (i+h  —  1)3(1  +  h  +  2)2(7  -f-  7&  -f  5)  ==  h*(3  +  7^2(12  -f  7/1)  is  -f  . 

Hence  at  x  =  1,  --  changes  sign  from  —  to  -(->  and  there  is  a  minimum  at  this  value. 
dx 

This  minimum  is  y  =  0. 

Finally,  to  test  x  =—  2,  -|-  =  (—  2  —  &  —  1)3(—  2  —  -  ft  -f  2)2(—  14  —  7h  +  5)  = 

(—  3  —  &)3(_.  fc)2(_  9  —  7fc),  which  is  -f.     Again,  ?j-  —  (—  2  -f  7i  —  1)3 


(—  2  +  7i  4-  2)2(—  14  -f-  7A  -f-  5)  =  (—  3  +  fc)3(-f-  A)2(_  9  -f  7/i)  is  also  +. 
Therefore  x  =  —  2  does  not  correspond  either  to  a  maximum  or  a  minimum. 

156.  Sen.  —  It  is  usually  easy  to  see,  without  going  through  with  the  de 
tails  of  the  substitution,  whether  -j  changes  sign  with  h  in  such  cases  as 

the  above  ;  that  is,  whether  if  x  =  a  is  the  critical  value  we  are  testing,  -~ 

will  have  a  different  sign  when  we  substitute  a  -f-  h  for  x,  from  what  it  will 
when  we  substitute  a  —  k  f  or  x. 

Ex.  11.  Examine  y  =  -  —        '   for  maxima  and  minima. 

(x  —  d)2 

13)  =  0,  gives  for  the  critical  values  x  =  —  2,  x  =  13. 


dx  (x  —  3 

*  =  £±W!L^a  =  oo,  giTCS  (x  -  ay  =  O,  whence  ,  =  3. 

dx  (x  —  *^) 

d-ii 
In  this  case  it  is  better  not  to  form  •=•=•  as  it  is  complicated,  but  test  the  critical 

values  by  noticing  whether  --  changes   sign  or  not  for  these  points,     x  =  —  2 

does  not  correspond  to  either  a  maximum  or  a  minimum,     x  =  13,  gives  y  =  33|, 
a  minimum,     x  =  3,  gives  y  =  oo,  a  maximum. 


Son.  —  The  first  10  examples  give  x  =  oo  for  -j-  =  oo,  and  hence  give 


rise 


to  no  critical  values,  as  x  =  oo  cannot  correspond  to  a  maximum  or  mini 
mum,  there  being  no  succeeding  value  of  the  function. 

Ex.  12.  Examine  y  —  —.  --  -—  for  maxima  and  minima. 

SUG'S.  —  Putting  --  =  0,  gives  x  =  1,  and  5,  as  the  critical  values.     Putting 

—  =  oc,  gives  x  =  —  1.     When  x  =  1,  y  —  Q,  a  minimum.     When  x  —  5,  y  =  -fo, 
a  maximum.     When  x  =•  —  1,  y  is  neither  a  maximum  nor  a  minimum. 


96  APPLICATIONS   OF   THE   DIFFERENTIAL   CALCULUS. 

Ex.  13.  Examine  y  =  b  -f  (x  —  a^  for  maxima  and  minima. 
Result,  The  critical  value  of  x  is  x  =  a.     But  this  does  not  corres 

pond  to  either  a  maximum  or  a  minimum,  since  —  does  not 

ax 

change  sign  at  this  value. 

d2?/ 
SUG.  —  In  this  example  —  =  +  oo  for  x  —  a,  and  hence  cannot  be  used  to  dis 

criminate  between  maxima  and  minima. 

Ex.  14.  Examine  y  =  b  -{-  (x  —  a)'3  for  maxima  and  minima. 

Result,  y  =  b  is  a  minimum. 

Ex.  15.  Examine  y  =  b  —  (x  —  aj'5  for  maxima  and  minima. 

Result,  y  =  b  is  a  maximum. 

Ex.  16.  Show  that  y  =  x3  —  3ar9  -f  Go:  -f-  7  has  neither  a  maximum 
nor  a  minimum  value. 

• 

Ex.  17.  Show  that  y  =  -  -  —  -  is  a  maximum  when  x  =  cos  jr.* 
1  +  &  tan  x 


-I-       anx/  >  >  COSX' 

ftji 

—  r       lo          . 

dc 

Ex.  18.  Show  that  y  =  sin3  #  cos  a?  is  a  maximum  when  ar  =  60°. 

CT  |-j    /-y* 

Ex.  19.  Show  that  y  =  --  -  —  is  a  maximum  when  x  =  45°. 
i  +  tail  a? 


GEOMETRICAL    PROBLEMS. 

Ex.  1.  Required  the  altitude  of  the  maximum  cylinder  which  can 
be  inscribed  in  a  given  right  cone  with  a  circular  base. 

SOLUTION.— Let  SO  =  a  be  the  altitude,  and  AO  =  b  the  radius  of  the  base 
of  the  given  cone.  Let  ac  =  x  be  the  altitude,  and  cO  =  of  —  y  be  the  radius 
of  the  base  of  the  required  cylinder.  The  function  which  is  to  be  a  maximum  is  the 
volume  of  the  cylinder.  Calling  this  V,  we  have  V  =  nipx.  In  this  form  V  is  a 
function  of  two  variables  x  and  y.  But  these  variables  being  dependent  upon 
each  other,  we  can  find  the  value  of  one  in  terms  of  the  other.  Thus,  S/  :  S  O  : : 

af  :  AO  ;  or,  in  the  notation,  a  —  x  :  a  : :  y  :  6  ;  whence  y  =  -(a  —  x).     Substi- 
*  When  x  =coax,x=  42"  21*  nearly. 


GEOMETRICAL  PROBLEMS. 


97 


tnting  this  value  of  y,  we  have  V  =  — r-va  — 
which  is  to  be  a  maximum.     Dropping  the  con 
stant  factor  ~  (153,  1st),  we  have  V—  (a  —  x)*x 

whence  x  =  ^a  ;  that  is,  the  axis  of  the  cylinder  is 
i  of  the  axis  of  the  cone.  From  this  we  readily 
find  y,  the  radius  of  the  base  of  the  cylinder  =  |6. 
.  • .  volume  of  cylinder  =  -^-Ttab'2.  But  volume  of 
cone  =  $7tab'2 ;  whence  volume  of  cylinder  =  $ 
volume  of  cone. 


FIG.  25. 


Ex.  2.  To  find  the  axis  of  the  maximum  cone  which  can  be  inscribed 
in  a  given  sphere. 

SUG'S. — Let  AmB  be  the  semicircle  which,  re 
volved  upon  A  B,  generates  the  sphere,  and  Aab  the 
triangle  which  generates  the  cone.  Let  A  O  =  r, 
A5  =  x,  a,ndab=y.  Then  V  =  ijr?/2;c=i7rx2(2r — x}, 
since  ab  =  y^  =  A&  X  b  B  =  x(2r  —  x).  .  • .  x  =  §r, 
or  the  altitude  of  the  cone  is  I  of  the  diameter  of 
the  sphere.  Volume  of  sphere  =  far3,  volume  of 
maximum  cone  =  28f  X  far 3 ;  or  the  cone  =  2\  of  the  sphere. 


SCH. — In  attempting  the  solution  of  such  problems,  first  notice  what  the 
function  is  which  is  to  be  a  maximum.  Thus,  in  Ex.  1,  it  is  the  volume  of 
a  cylinder  ;  in  Ex.  2,  it  is  the  volume  of  a  cone.  Having  obtained  the  equa 
tion  expressing  the  function  in  terms  of  the  variable  or  variables  on  which 
it  depends,  if  there  are  two  dependent  variables  involved,  find  from  the 
conditions  of  the  problem  the  relation  between  these  variables,  and  sub 
stitute  for  one  of  them  its  value  in  terms  of  the  other.  Finally,  we  have 
a  function  of  a  single  variable,  which  can  be  examined  for  maxima  and 
minima  values  in  the  usual  way. 

Ex.  3.  Required  the  cylinder  of  greatest  convex  surface  which  can 
be  inscribed  in  a  given  right  cone  with  a  circular  base. 

SUG. — The  function  is  the  convex  surface  of  a  cylinder.  Using  the  same  notation 
as  in  MK.  1,  and  letting  S  represent  the  function,  we  have  S  =  lityx  = 

—  (a  —  x)x.     .'.  x  =  ia,  and  S  —  -5-  ;  that  is,  the  altitude  of  the  cylinder  is  i 
that  of  the  cone  ;  and  the  convex  surface  of  the  cylinder  is  to  the  convex  surface 

of  the  cone  as  -  :  v/a2  -f~  b'2,  or  as  £  the  altitude  of  the  cone  is  to  its  slant  height. 
a 


Ex.  4.  Required  the  maximum  cylinder  which  can  be  cut  from  a 


98 


APPLICATIONS   OF   THE   DIFFERENTIAL   CALCULUS. 


given  sphere.     The  axis  of  the  cylinder  =  f  v/3  times  radius  of  sphera 

The  cylinder  is  to  the  sphere  as  • — ~ :  1. 

V3 

Ex.  5.  Required  the  area  of  the  greatest  rectangle  which  can  be 
inscribed  in  a  given  circle. 

The  rectangle  is  a  square,  and  its  area  =  2r2. 

Ex.  6.  "What  is  the  altitude  of  the  maxi 
mum  rectangle  which  can  be  inscribed  in  a 
given  parabola  ? 

SUG'S.— Let  ac  =  x,  qf  =  y,  and  AX  =  a.  Let 
A  be  the  function,  the  area  of  the  rectangle.  Then 
A  '=  2a-?/.  From  the  equation  of  the  parabola 
^f  =  2p  X  A/,  or  2/2  ==  2p(a  —  x)  ;  whence  A  = 
2xv/2pva  —  aij.  A'  =  ax2  —  x3,  and  x  =  fa. 

Ex.  7.  Required  the  axis  of  the  cone  of  maximum  convex  surface 
which  can  be  inscribed  in  a  given  sphere. 

The  axis  =  $  the  radius  of  the  sphere. 

Ex.  8.  Required  the  altitude  of  the  maximum  cone  which  can  be 
inscribed  in  a  given  paraboloid,  the  vertex 
of  the  cone  being  at  the  intersection  of  the 
axis  of  the  paraboloid  with  the  base. 

SUG'S. — Let  ABC  be  the  parabola  whose  revo 
lution  about  AS  as  an  axis  generates  the  parabo 
loid.  Let  AS  =  a  the  axis  of  the  paraboloid, 
oS  —  x,  the  altitude  of  the  cone,  and  ao  =  y  the 
radius  of  the  base  of  the  cone.  The  result  is 
x  =  |ct. 

Ex.  9.  Required  the  maximum  para 
bola  which  can  be  cut  from  a  right  cone 
with  a  circular  base,  knowing  that  the 
area  of  a  parabola  whose  limiting  co-ordi 
nates  are  x  and  y  is  $xy. 


FIG.  28. 


SUG'S.— Let  SO=«,  BO  =  b,  AX  = 

aX  =  y-     The  function  is  A  (the  area) 


•x,  and 


But  aX  =  y  =  v/BX   X   XC  ;    and  CX   : 

C  B  :  AX  :  S  B,  or  CX  :  26  :  :  x  :  \/a*  -f-  l>n~ ; 
whence  CX  =  -J-,  letting  S  =  v/aa  -f  62  for 

brevity.     Theri  BX  =  C  B  —  CX  =  26  - 


FIG.  29. 


GEOMETRICAL   PROBLEMS. 


99 


=  -rr(S  —  x).     Finally,  A  =  $x      ~x(S  —  x)  =  — \/a;:!(,S'  —  x),  and  A' 


—  x4.     The  result  is  x  —  £8,  that  is,  the  axis  of  the  parabola  is  |  the  slant  height 
of  the  cone.     The  area  of  the  parabola  =  ibS\/3.     Notice  that  CX  =  |CB. 

Ex.  10.  From  a  given  quantity  of  material  a  cylindrical  vessel  with 
circular  base  and  open  top  is  to  be  made,  so  as  to  contain  the  greatest 
amount.  What  must  be  its  proportions  ? 

SUG'S. — Let  a;  =  the  altitude,  y  the  radius  of  the  base,   and   V  the  volume. 

Then   V  =  TCV*X  is  to  be  a  maximum.     Hence   —  =  %vx—  4-  w2  =  0    or  w  == 

dx          y  dx 
flu 

But  Z-jtyx  4-  7tyz  =  s,  the  surface.     Differentiating  %7txdy  -f  Zicydx  4- 


Znydy  =  0  ;  whence  --  = 

dx  x  + 


Substituting,  y  = 


.-.  y  =  x,  that 


is,  the  altitude  =  the  radius  of  the  base.     The  altitude 


II. 

\J37T 


Ex.  11.  Of  all  right  cones  of  a  given  convex  surface  to  determine 
that  whose  solidity  is  the  greatest. 

The  altitude  =  \/2  into  the  radius  of  the  base. 
Ex.  12.  To  find  the  maximum  rectangle  inscribed  in  a  given  ellipse. 

c.     >                                             dA'  dy 

DUG  s.     A.  =  4Lxy.     A  =  xy.     =  y  -\-  x--  —  ( 

.  dv  _       B?a;  dy 


dx  A-y  "dx         A-y' 

x:y::A:B.  That  is,  the  sides  of  the  rectangle  are  to  each 
other  as  the  axes  of  the  ellipse.  The  sides  of  the  rectangle 
are  A\/%,  and  B\/%. 


FIG.  30. 


Ex.  13.  To  find  the  maximum  cylinder  which  can  be  inscribed  in  a 
given  ellipsoid,  generated  by  the  revolution  of  an  ellipse  about  its 


transverse  axis. 


The  axis  of  the  cylinder  =  — =A. 


Ex.  14.  A  person  being  in  a  boat  3  miles  from  the  nearest  point  of 
the  beach,  wishes  to  reach  in  the  shortest  time  a  place  5  miles  from 
that  point  along  the  shore  ;  supposing  he  can  walk  5  miles  an  hour, 
but  pull  only  at  the  rate  of  4  miles  an  hour,  p  x  A 

required  the  place  where  he  must  land. 


SUG'S.  —  Let  AX  =x,  and  £  =  the  time  required  to 
reach  A  by  rowing  from  B  to  X,  and  walking  from 


x    .    v/i5  — 
X  to  A.     <  =  -  -|  --  - 


4-  9 


. 

is  to  be  a  mini 


mum.     He  must  land  at  X,  1  mile  from  A. 


FIG.  31. 


100  APPLICATIONS   OF   THE  DIFFERENTIAL  CALCULUS. 

Ex.  15.  Divide  a  into  two  factors  the  sum  of  which  shall  be  a  min 
imum.  Result,  The  factors  are  equal. 

Ex.  16.  The  difference  between  two  numbers  is  a ;  required  that 
the  square  of  the  greater  divided  by  the  less  shall  be  a  minimum. 

Result,  The  greater  =  twice  the  less. 

Ex.  17.  To  find  the  number  of  equal  parts  into  which  a  must  be 
divided,  so  that  their  continued  product  shall  be  a  maximum. 

(Q\X 
-J  .     logw  =  x(loga —  logic),     u    =  xloga  — 

xlogx.     —  =  log  a  —  logo;  —  1  =  0.     x  =  ~.     Arithmetically  the  problem  is 
possible  only  when  —  is  integral. 

Ex.  18.  Find  a  number  x  such  that  its  #th  root  shall  be  a  maxi 
mum.  •  x  =  e. 

Ex.  19.  A  privateer  wishes  to  get  to  sea  unobserved,  but  has  to  pass 
between  two  lights,  A  and  B,  on  opposite  headlands,  the  distance  be 
tween  which  is  a.  The  intensity,  at  a  unit's  distance,  of  A  is  6,  and 
of  B,  c.  At  what  point  must  the  privateer  cross  the  line  joining  the 
lights,  so  as  to  be  as  little  in  the  Ijght  as  possible  ;  it  being  under 
stood  that  the  intensity  of  a  light  at  any  point  equals  its  intensity 
at  a  unit's  distance  divided  by  the  square  of  the  distance  from  the 
light. 

b  f- 

SUG. — Letting  x  =  the  distance  from  A,  the  function  is  u  =  —  -j- 


x*   '    (a  —  x)2 


»*+.* 

Ex.  20.  The  intensity  of  illumination  from  a  given  light  varies  as 
the  sine  of  the  angle  under  which  the  light  strikes  the  illuminated 
surface,  divided  by  the  square  of  its  distance  from  the  surface.  Re 
quired  the  height  of  a  light  directly  over  the  centre  of  a  given  circle, 
so  that  it  shall  illuminate  the  circumference  as  much  as  possible. 

SUG'S. — Let  /represent  the  illumination  at  P,  which  is  to  be  a 
maximum  ;   PO  =  R  ;  and  LO  =  x.     I  =  —          — . 


jRa  -f.  x*  —  So;*  =  0,  and  x  —  JR\/  J. 


GEOMETRICAL  PROBLEMS.  101 

Ex.  21.  To  find  in  a  line  joining  the  centres  of  two  spheres,  the 
point  from  which  the  greatest  portion  of  spherical  surface  is  visible. 


Suo's. — The  function  is  the  sum  of  the  two  C. 


zones  whose  altitudes  are  M  D  and  md  : 
hence  we  must  obtain  an  expression  for  the 
areas  of  these  zones.  Let  CO  =  R,  co  =  r, 


V 


\ 


Oo  =  a,  PO  =x,  and  Po  =  *'=«-*.     °    a 

From  the  right  angled  triangle   PCO,  R*  =  FIG.  33. 

DO  X  x  ;  whence  MO  =  R  —  DO—  —        —  ,  the  altitude  of  the  zone  seen 

A*rt»'       _      A*2 

on  this  sphere.     In  like  manner  md  =  --  ;  —  .     Now  the  area  of  a  zone  being  to 
the  area  of  the  surface  of  its  sphere  as  the  altitude  of  the  zone  is  to  the  diameter 

7~j  7^2 

of  the  sphere,  letting  Z  and  z  be  the  zones,  Z  :  ±7tR*  ::  —          —  :2R,.-.Z  = 
_J?.     And  in  like  manner  z  =  lit™  ~  ** 


Hence,    letting    S   represent    the    function,   we    have,    S  =  ZxR*—  -\- 


•C 


,  S'  =  B.  -       +  r«  --  r—.          =-  --         =  0  ;  whence 

^  *  —     * 


a  —  x  x    '  a  —  x     ax        x-        (a  — : 

a          3. 

aR                                                                            (r'z    \   R'2)2~\ 
x  =  — ;  and  the  entire  surface  =  2it\  rz  -{-  Rz . 

Since  27Tr2  -f-  27fJ22  is  the  sum  of  the  hemispheres,  8  is  always  less  than  this  sum 
except  when  0=00. 


GENERAL   SCHOLIUM. 

The  student  should  now  resume  the  study  of  General  Geometry  at  Chap 
ter  IV. 


CHAPTER  III, 

THE  INTEGRAL    CALCULUS. 


SECTION  I. 

Definitions  and  Elementary  Forms, 

The  Integral  Calculus  is  that  branch  of  the  Infinites 
imal  Calculus  which  treats  of  the  methods  of  deducing  the  relations 
between  finite  values  of  variables,  from  given  relations  between  the 
contemporaneous  infinitesimal  elements  of  those  variables.  It  is  the 
inverse  of  the  Differential  Calculus. 

158.  The  Integral  of  a  differential  function  is  another  func 
tion  which  being  differentiated  produces  the  differential. 


(.  Integration  is  the  process  of  deducing  the  integral  func 
tion  from  its  differential. 

160.  The  Sign  of  Integration  is  f,  which  is  a  form  derived 
from  the  old,  or  long  s.  It  is  the  initial  of  the  word  sum,  and  came 
into  use  from  the  conception  that  integration  is  a  process  of  summing 
an  infinite  series  of  infinitesimals. 

ILL'S. — Suppose  we  have  given  dy  =  — — — -— .     This  is  a  differential  function, 

and  we  have  given  in  the  equation  the  relation  between  dy  and  dx.  The  Integral 
Calculus  proposes  to  find  the  relation  between  y  and  x  from  such  a  relation  between 
their  differentials  ;  or,  in  other  words,  to  find  the  function  which  being  differen 
tiated  produces  the  given  differential.  The  function  in  this  case  is  y  =  — - — -,  as 

will  be  proved  by  differentiating.     The  latter  is  therefore  called  the  integral  of  the 

r-           C    4xdx              2a*2 
former.     Using  the  sign  of  integration,  we  may  write  J  dy  =  I  — — 2  = ; 

and  read,    "the   integral  of  dy  equals   the  integral  of    - —      — ,  which   equals 


The  conception  of  integration  as  a  process  having  for 
its  object  the  summation  of  an  infinite  series  of  infinites 
imals  may  be  illustrated  by  considering  the  area  of  an 
ellipse  as  composed  of  an  infinite  number  of  infinitesi 
mal  segments,  as  represented  in  the  figure.  Let  A  rep 
resent  the  area  of  the  ellipse  ;  whence  d  A  represents  one 
of  the  infinitesimal  segments,  or  elements  of  the  area. 


FIG.  34 


DEFINITIONS   AND   ELEMENTARY   FORMS.  103 


Now  it  is  found  that  (7 A  =  -(a9  —  0»)\te     By  integration  it  is  found  that  the 

entire  area  is  nab,  a  and  b  being  the  semi-axes.  But,  as  the  entire  area  is  the  sum 
of  the  infinitesimal  segments,  the  process  of  integration  may  be  considered  as 
having  for  its  object  the  summing,  or  adding  together  of  all  the  infinitesimals 
which  go  to  make  up  the  entire  area. 

161.  IMPORTANT  GENERAL  STATEMENT. —Strictly  speaking,  there  is 
no  such  thing  as  a  Process  of  Integration.  Whenever  a  differential  is 
proposed  for  integration,  the  first  question  is,  Is  this  a  Known  Form  ? 
that  is,  Can  ice  see  by  inspection  what  function,  being  differentiated,  pro- 
daces  this?  If  we  cannot  thus  discern  the  integral  by  a  simple  inspec 
tion,  the  only  question  remaining  is,  Can  we  transform  the  differential 
into  an  equivalent  expression  the  integral  of  which  we  can  recognize? 
Thus,  in  any  case,  we  pass  from  the  differential  to  its  integral  by  a 
simple  inspection  ;  and  the  sufficient  reason  always  is,  This  expression 
is  the  integral  of  thai,  because,  being  differentiated,  it  produces  it. 


THREE   ELEMENTARY    PROPOSITIONS. 

162.  Prop.  1.—  Constant  factors  or  divisors  appear  in  the  integral 
the  same  as  in  the  differential,  and  hence  may  be  written  before  or  after 
the  sign  of  integration  at  pleasure. 

DEM.—  This  is  a  direct  consequence  of  the  fact  that  constant  factors  or  divisors 
appear  in  the  differential  the  same  as  in  the  function  (48). 

163.  Prop.  2.—  To  integrate  the  algebraic  sum  of  several  differen 
tials,  integrate  each  term  separately,  and  connect  the  integrals  by  the  same 
signs  as  their  differentials  were  connected. 

DEM.—  This  is  a  direct  consequence  of  the  rule  for  differentiating  the  algebraic 
sum  of  several  variables  (51). 

164.  Prop.  3.—  An  indeterminate  constant  must  always  be  added 
to  the  integral  of  a  function. 

DEM.  —Since,  in  differentiating,  constant  terms  disappear,  in  returning  from  the 
differential  to  the  integral  we  have  to  represent  any  possible  constant  terms  by  an 
indeterminate  constant. 

gCH  _The  method  of  disposing  of  this  constant  term,  which  we  usually 
represent  by  C,  will  be  presented  hereafter.*  The  fact  that  there  may  be 
such  a  term  is  all  that  the  student  is  expected  to  see  at  this  point.  To  illus 
trate,  suppose  y  =  3rt*»  +  126,  dy  =  Gax  dx.  Now,  if  the  latter  alone  were 
given,  we  might  see  that  y  =  Sax*  was  its  integral,  since  being  differentiated 
it  would  produce  dy  =  Qax  dx.  But  so  will  y  =  a&  +  any  constant,  as  126, 
or,  as  we  represent  it,  y  =  3nxz  -}-  C.  _  ^^^ 


*  Section  VII.,  closing  illustration. 


THE  INTEGRAL  CALCULUS. 

TWO   ELEMENTARY  RULES. 

165.  RULE  1.  —  Whenever  a  differential  can  be  separated  or  trans 
formed  into  three  factors  ;  viz.,  1st.  Its  constant  factors  ;  2nd.  A  vari 
able  factor  affected  loith  any  exponent  except  —  1  ;  and  3rd.  A  differen 
tial  factor  which  is  the  differential  of  the  2nd  factor  without  its  exponent, 
its  integral  is 

THE  PKODUCT  OF  THE  SECOND  FACTOR  WITH  ITS  EXPONENT  INCREASED  BY  1, 
INTO  THE  1ST  OR  CONSTANT  FACTOR  DIVIDED  BY  THE  NEW  EXPONENT.* 

DEM.—  This  rule  is  evident  from  (162),  and  the  rule  for  differentiating  a  variable 
affected  with  an  exponent  (56).  Thus,  if  y  =  m[/(a?)]«,  dy  =  mw[/(o?)]—i  d  ./(*), 
or  mn  X  [/(tf)]"-1  X  d.f(x)  ;  whence  to  pass  from  the  latter  to  the  former,  we  have 
to  suppress  the  differential  factor,  d.f(x),  increase  the  exponent  n  —  1  by  1  making 
it  n,  and  divide  the  constant  factor  mn  by  this  n. 

In  the  exceptional  case  the  exponent  by  which  we  would  be  required  to  divide 
according  to  the  rule  would  be  1  —  1  =  0,  whence  the  result  would  be  oo. 

Ex.  1.  Integrate  dy  =  3ax*dx. 

SOLUTION,     dy  =  3a  X  »2  X  dx  ;  whence  y  =  fsax*dx  =  ~x*  -f  C  =  ax3  -f-  C. 

Ex.  2.  Integrate  dy  =  ax3dx. 

SOLUTION,     dy  =  a  X  x3  X  dx.     .'.  y=  fax^dx  =  ic4  -f  G. 

Ex.  3.  Integrate  dy  =  (a  -f-  3x*)*6xdx. 

SOLUTION,  dy  =  1  X  (a  -f-  3X*)*  X  6xdx,  which  corresponds  to  the  requirements 
of  the  rule,  since  d(a  -f  3a?)  =  Qxdx.  .  •  .  y  =  J(a  +  3&)*Gxdx  =  i(a 

Ex.  4.  Integrate  dy  —  (a  -f 


SOLUTION.  —  The  differential  of  the  quantity  within  the  parenthesis  being  Qxdx, 
we  write  dy  =  ^x(a  -f  3o?*)3  X  §xdx,  which  conforms  to  the  requirements  of  the 
rule.  .•  .  y  =  J"|f(a  -f-  SxPp&cdx  =  ^-(a  +  3^)4  -f  a 


Ex.  5.   Integrate  dy  =  a(ax 

SUG'S.    y=^[a(a»+6a 

=  /[I  X  (aa?  +  bx*)*  X  (a  4-  2&^)dx]  =  |(aa;  -f-  6x2)^  4.  (7. 

.i#{>.  RULE  2.  —  WHENEVER  A  DIFFERENTIAL   CAN  BE  WRITTEN  IN,  OR 

TRANSFORMED  INTO  A  FRACTION  WHOSE  NUMERATOR  IS  THE  EXACT  DIFFEREN 
TIAL  OF  ITS  DENOMINATOR,  THE  INTEGRAL  IS  THE  NAPIERIAN  LOGARITHM  OF 
THE  DENOMINATOR.* 

*  In  giving  such  rules  the  constant  term  of  the  integral  is  not  mentioned,  as  its  addition  is 
always  implied. 


DEFINITIONS  AND   ELEMENTAKY  FORMS.  105 

DEM. — This  is  a  direct  consequence  of  the  rule  that  the  differential  of  the  Na 
pierian  logarithm  of  a  number  is  the  differential  of  the  number  divided  by  the 
number.  [This  will  be  seen  to  be  the  exceptional  case  under  the  preceding  rule.  ] 


167.    ELEMENTARY  FORMS. 

1.  V  =   fxndx  =    r*n+1  +  C.     Same  as  Rule  1. 

J  n  -f  1 

2.  y  =  C—  =  log*  +  G.     Same  as  ^wZe  2. 

J    & 

3.  y  =   /Vd!a?  =  ^— a1  +  (7.  Converse  of  (60). 

J  log  a 

3X.  y  ==  JVtfa?  =  (?  +  a  "          (61). 

4.  y  =  fcosxdx  =  sin*  +  G.  (66). 

5.  y  =  f  -—  sin*J*  =  cos*  -f  01  (67). 

6.  y  =   T— -,  or  fsec^*^*  =  tan*  +  C.  "         (69). 

J  cos2*        J 

7.  y  =  T ^-,  or  f  —  cosecs*  ^*  =  cot*  +  01      "          (70). 

J        sin2*        J 

8.  y  =  ftan  *  sec  *  dx  =  sec  *  +  0.  (71)' 

9.  y  =  f —  cot*cosec*d*  =  cosec*  +  G.  (72). 

10.  y  =  fsmxdx  =  vers*  +  C.  "          (73). 

11.  y  =  jf  —  cos  *  d*  =  covers  *  -f  01  (74). 

12.  y  =  C—^—  =  sin-1*  +  C.  «          (75). 

(77). 

(79). 
"  (80). 

(81). 
"  (82). 

(83). 

(84). 


.  y  =   C X         = 

»/  *V   *2   1 


,  y  =  /" jr         = 

^        v/2*  —  ** 


106  THE  INTEGRAL  CALCULUS. 

168.    SUBORDINATE    CIRCULAR   FORMS. 

r        dx  1    ,        bx 

1.  y  =  /  -;==  =  -  sin-1  -  +  (7, 
./         '2  —    28        o  a 


a'2  — 

or  =   /          "-  --  =  sin"1-  +  C,  when  6  =  1. 
./    v  aa  —  ar*  a 

/"  dx  1  6a? 

2.  y  =  /  --  -  --  —  =  -cos-1    -  -f  (7, 
./         v/fli  _  ASJ;*        b  a 


/*             d  T,  x 

or  =   /  -    — =  cos"1  -  -f  (7,  when  6=1. 

;.  y  =    / -, —  =  —  tan"1  —  +  ft 

J  a'2  -f  62^        a6  a 

or  =    C    dx      =  -  tan-1-  +  (7,  when  6=1. 
J  a*  +  x*       a          a 

f  dx  1  bx 

--  y  =  I  -  -  -r-r-r:-,  =  ~  cot-1    -  +  G, 


or 


a3  +  te2        ab 

=   I a-—  =  -  cot"1  -  +  <7,  when  6=1. 

J        a2  +  a;2        a  a 


5.  y=    I       _f! =  1  sec-1  ~  +  ft 

«/  •&  v  6*37*  —  a2        a 

or  =    /  — - — '- =  -  sec"1  -  +  ft  when  6  =  1. 

•/  57V  ^2  —  a2        a  a 

/•  </JT  1  ,6.r 

6.  y  =  / -; —        _  =  -  cosec"1 f  (7, 

^        xvfcx*  —  a*        a  a 

/6?jr  1  .r 
; —      =  -  cosec"1  -  +  C,  when  6=1. 

7.  y  =    C  dx  =  1  vers-1-  +  (7, 

/r/jr  ^r 
=  vers    — \-  C,  when  6  =  1. 
v2ax  —  x*  a 

8.  y  =   I  -  - — • —  =  -  covers"1 \-  ft 

C  dx  vX        ^ 

or  ==  I =  covers     -,  when  6=1. 


DEM.— These  forms  maybe  considered  as  the  converse  of  Ex's.  1,2,  pages  38,  39. 
They  may  also  be  established  by  differentiating  the  result  and  showing  that 

its    differential   is    the    given    differential    function.       Thus, 


DEFINITIONS   AND   ELEMENTARY  FORMS.  107 

b  , 


.  ...          .. 

jl  _  ^!f!  |a*  ~  62x8 

\j  «-  \        a- 


vV  — 


bd.r  dx 

=  .       [The  student 


should  verify  all  of  them  in  this  way.] 

^4  direct  way  of  obtaining  these  integrals,  and  one  with  which  the 
student  should  not  fail  to  become  familiar,  is  the  following  : 

To  integrate  dy  =  /  -  "         ,  we  observe  that  it  has  the  general  form  of  the 


=  /  -  " 
J    v/«2  —  b*x* 


differential  of  an  arc  in  terms  of  its  sine,  which  is  —  ''  •     -.     To  transform  our 

v/1  —  *'* 
expression  into  this  form,  we  have  first  to  make  the  first  term  under  the  radical  1. 

This  can  be  readily  done  thus,    C        d*    -  =      /^ ^          =  1     C     '*'' 

J    Va*  —  6x2         /        |        b*x*      a  j      i        b-x* 
/a      1 /        1  —  — 

J  \     <*    J  \ 

since  the  constant  divisor  a  appears  in  the  same  form  in  the  integiui  as  in  the  dif 
ferential  (102).  Now  to  make  the  quantity  under  the  sign  of  integration  the 
differential  of  an  arc  in  terms  of  its  sine,  the  numerator  ought  to  be  the  differen 
tial  of  the  square  root  of  the  second  term  in  the  denominator,  which  is  the  square 

of  the  sine.     But  d(  —  )  —  -dx.    We,  therefore,  need  to  introduce  -  into  the  nu- 
\  a  /       a  a 

merator.     This  can  be  done  by  putting  -  outside  the  sign  of  integration  as  they 

C  r    -dx 

will  neutralize  each  other  (162).     Hence  -  /  —      *        =  1  .  -  /          a         = 

a  j  62«2       a    b  /  ^2X2 

/         1— __  /         l_  -- 

*s     \j          a2  +J     \l          a2 

f^       ~dx 

1    /      .  — -     The  quantity  now  under  the  sign  of  integration  is  the  exact 

0  /  b*x* 

J    NJ1         ~& 

differential  of  sin-'—,  since  it  is  the  differential  of  the  sine,  -,  divided  by  the 
a  a 

square  root  of  1  — the  square  of  the  sine,  — (75).      Hence,  finally,  as  J  dy=y, 

S*         dx  1    .        bx 

we  have  y  —    I =  -  sm-i p-  C. 

i    \/ eft b^x^  ® 

[The  student  should  produce  all  these  subordinate  integrals  in  this  way,  for  the 
benefit  of  the  exercise.  We  give  the  outline  of  two  more,  which  should  be  ex 
plained  at  length  as  above.  ] 

r  r  r  -<& 

dx  I          dx  _  /      dx  I     a    I      a 

y«<i+^-)  t/i+^7  a°  i/i+b-^ 


108  THE  INTEGRAL  CALCULUS. 

dx         r*      dx       1 


dx 

~  =  covers—1  -  -f-  C. 


1          C  a r  a 

a  '  a  /  —^  ~      I 

J   J«(i)-5  7   vKi)-S 


169.    LOGARITHMIC   TRIGONOMETRICAL   FORMS. 

^_dx 
dx  r  r  dx 


,  r  dx  r  r  d 

1.    y   =    /  -^  —    or    J  cosec  xda;   =    /  —  —  ; 

J  sinx  J  %sin(lx)coB(ix) 


tan  (£#) 


_ 


2.    y  =  f*L  or  /«o«fc  =   /V-£  -  =  - 
J  cosa;       J  J  sm(i7T  —  x) 


—  a;) 
C. 
dx          f     ,  /*cos 


=  [by  (1)] 
L  J 


/*  dx          f     ,  /*cos  xd#          /*d(sin  x) 

3.  y  =    /  -  --  or  J  cot  xdx  =    /  -    --  =    I  -  -  -  '  =  log  sin  x  +  C. 

J  t&nx      J  J      smx          ^/      smx 

/*  dx  /-,  /*sin  xdx  /*<i(cos  a;) 

4.  w  =    /  -  or  I  tan  xdx  =    /  -  =  —  I  —  -  '  =  —  log  cos  x  = 

I  cot  x       J  J     cos  x  J      cos  x 


5.    y  .   f^—  =  f^-  =    /*4^ 
J  smx  cos  a;      J  sm(2x)       J  sin  (2s 


[by  (1)]  logtan. 


.  —  The  above  32  forms  must  be  so  thoroughly  memorized  as  to  be  in 
stantly  recognized.  There  is  no  doing  anything  in  the  integral  calculus 
without  this.  These  forms  are  to  integration  what  the  multiplication  table 
is  to  arithmetical  operations.  Thus  we  say,  7  goes  into  56  8  times,  because 
8  times  7  =  56.  In  like  manner  we  say  that  cottar  -f  C  is  the  integral  of 

-,  because  cot"1  a;  -J-  C  differentiated  =  —  -  —  -'-  —  . 


Ex.  1.  Integrate  dy  =  3ax3dx. 

SOLUTION. — The  integral  of  dy  is  y,  since  y  differentiated  =  dy.     To  integrate 
x'dx,  notice  that  3a  X  tf3  X  dx,  conforms  to  (165).    .'.  y  =  J~3ax3dx  = 

>  +  a 

Ex.  2.  Integrate  dy  =  (2a  -f  3bx)*dx. 

SITG'S.     y  =  J*(2a  -f  3bx)*dx  =  J"(8a3  -f  36a26x  -f  54a&2x2-f  2763x3)dx  = 


DEFINITIONS   AND   ELEMENTAKY  FORMS.  109 


-f  J"36a26xdx  -f  J"5±ab*aflda;  +  J"2763#3dx  =  8a3x  +  18a2&x*  -f 
-f  YW  +  C  (163). 
This  may  also  be  integrated  by  (165).     Thus  y  =  C     X  (2o  -f  36x)3  X 


-^r(2a  -f-  3&x)4  -f  (7;  which  is  the  same  as  the  preceding. 


Ex.  3.  Integrate  dy  = 


SUG'S.     y  =  C      XdX  -  =  f  (a2  +  ««f  *«fa  =  /i  X  (a2  +  #2)~  ^  X  2xdx  =: 
J   Va*  +  x2 

+  ^J  +  a 

Ex.  4.  Integrate  dy  =  bx^doc.  y  =  %bx%  +  C. 

Ex.  5.  Integrate  dy  =  ^x^dx.  y  =  —  x~z  +  (7. 

Ex.  6.  Integrate  dy  =  2o:*da?.  y  =  f  ^  +  C. 

Ex.  7.  Integrate  dy  =  —  5mx~*dx.  y  =  —  ^-mx^  +  C. 
Ex.  8.  Integrate  dy  ==  -. 


Ex.  9.  Integrate  dy  =  —  r  y  =  f(a»  +  a;3)    +  C7. 

(a2  +  x^ 

Ex.  10.  Integrate  dy  =  -  —^dx. 

(Sax2  —  x3)* 

gUG's.    --  2ax  —  x*        _  _  ^axZ  _  x3)-3  (2ax  —  x2)dx  =  —  i  X  (3ax2  —  «3)~ 
(3ax2  —  a;3)3 

-2 

a-  =  _       ax  —  x3 
x3) 

Ex.  11.  Integrate  dy  =  I2bx(4bx*  —  2cx3fax  — 


/Q//7*  _  a«2  -2 

---  :  -  -da-  =  _  ^(3ax2  —  x3)3  +  G. 
(3ax2—  x3)3 


SUG'S.     12&x(46a;2  --  2cx3)cZx  —  9cx2(46x2  — 
(126x  _  9cx2)dx.     Now  in  order  that  the  factor  (12&#  —  9cx2)dx  should  be  the  dif 
ferential  of  46#2  —  2cx3  we  should  have  8  instead  of  12  and  6  instead  of  9.     Hence 

—  2cx3)3(86x  —  6cxa)cZx.     .  •  .  y  =  !(4&#2  —  2cx3)^  -f-  C. 


17O.  Son.  —  It  is  not  always  easy  to  determine  just  what  constant  factor 
is  required  in  order  to  make  the  differential  factor  the  differential  of  the 
quantity  within  the  parenthesis  ;  nor  can  such  a  factor  always  be  found. 
To  determine  whether  there  is  such  a  factor  or  not  ;  and,  if  there  is,  to 
find  it,  we  may  proceed  as  in  the  following  examples. 


110  THE  INTEGKAL  CALCULUS. 


2aa?  ~ 


Ex.  12.  Integrate  dy  =  -        ~       _  -dx. 
(26 


SUG>S-  —jdx  =  (26  +  Box*  —  5*)~*(Sta»  —  5z2)cfo.    Suppose  J.  to 

—  5x3)  3 


be  the  constant  factor  sought,  so  that  y= 


=  i- 


It  is  required  that  A  should  fulfill  the  condition  d(26  -}-  3o»8  —  5x3)  =  la  Ax—  5  Ax*, 
or  6ax  —  15x2  —  ZaAx  —  5  Ax*.  Now,  as  this  is  to  be  true  for  all  values  of  x,  we 
have  6a  =  2aA,  or  ^L  =  3  ;  and  also  15  =  5  A,  or  .4  =  3.  Hence  3  is  the  factor 
sought,  and  we  have 

y  =  f  1          *3  ~* 


i(26  +  Sax*  —  5o;3)*  _|_  a 
Ex.  13.  Integrate  dy  =  x(l  +  x2  —  Zx^dx  —  3^(1  +  x*  — 


SUG'S.    dy  = 

We  are  to  seek  a  factor  A,  which  fulfills  the  condition  d(l  -f  x2  _  2a;5)  = 
(^4#  —  3^a;4)dx,  or  performing  the  differentiation,  and  dropping  dx,  2x  —  10.r*  = 
Ax  —  3Ax*  ;  whence  the  first  condition  required  is  A  =  2,  and  the  second  is 
3  A  =  10,  or  A  =  *£-.  These  conditions  being  incompatible  with  each  other, 
there  is  no  factor  which  meets  the  conditions,  and  the  integration  cannot  be  per 
formed  in  the  manner  now  under  consideration. 

Ex.  14.  Integrate  dy  =  g  -  J)cte.  y  =  -  ^  +  I  +  C. 

Ex.  15.   Integrate  dy  =  (  f  ax%  —  f  bx*  )dx.        y  =  ax*  —  bx?  +  C. 

Ex.  16.  Integrate  dy  =  ax*dx  -\  --  '—.  y  =  -^-  -f  x?  +  C. 

2vx  3 

Ex.  17.  Integrate  dy  =  (2^4  —  3^2  +  1)~* 


y  = 

Ex.  18.  Which  of  the  following  can  be  integrated  by  the  method 
used  in  the  preceding  examples  ;  viz.,  dy  =  (5x2  —  2x)^(3x  —  5)dx  ; 
dy  =  5(5a?  —  Zx^xdx  —  (5&  —  Zx^dx  ;  dy  =  -  ^~1  flx  . 

(1  —  x  -j-  x*y 


Sen.  —  Caution.  The  student  must  not  fail  to  observe  that  it  is  only 
constant  factors  which  he  can  introduce  in  the  manner  illustrated  in  the 
preceding  examples.  When  the  differential  factor  is  not  of  the  right  form 


DEFINITIONS  AND   ELEMENTARY  FORMS.  Ill 

as  to  the  variable,  nothing  can  be  done  with  it  by  this  process.  Were  we 
to  attempt  to  introduce  a  variable  factor  into  the  differential  factor,  its  re 
ciprocal  would  have  to  be  introduced  into  the  1st,  or  constant  factor,  and 
this  would  destroy  the  condition  that  the  first  factor  is  constant.  It  is  always 
well,  if  there  is  the  least  doubt  whether  the  integral  found  is  correct,  to  differen 
tiate  it  and  see  if  it  gives  the  proposed  differential. 

171.  Prop. — It  is  sometimes  possible  to  bring  a  differential  to  the 
form  required  in  (165)  by  transposing  one  or  more  factors  of  the  va 
riable  from  the  factor  in  the  parenthesis  to  the  differential  factor,  or  vice 
versa. 

axdx 
Ex.  19.  Integrate  dy  =  -          -. 


SOLUTION.      i___    =    a(2&a;  +  x^^x    —    O(2bar-'  -f  l)~V«dflS    = 

(26x  +  a*)* 

__  ^L  x  (2bx~*  -f  1)~*  X  (—  2byr-*dx},  in  which  —  26ar~2dx  is  the  differential  of 

26 

(260J-1  +  1).     . ' .  y  =  C-^—z  =  T-  I  X  (2te->  +  1)~*  X  (- 
J     Zbxx          J 


Ex.  20.'  Integrate  dy  = 


" 


adx  a  -\  .    -  .   „ 

SUG.     -  -  =^-(36{g-i-f4ca)  23bx-*dx,  y  =  ----  ^—    --  h  C. 


Ex.  21.  Integrate  dy  =  ~(166).          y  =  log  (1  +  &)  +  0. 

SCH.  —  When  the  integral  is  a  logarithm,  it  is  customary  to  write  the 
constant  also  as  a  logarithm.  Thus,  in  the  above  example  if  we  let  log  c  =  C, 
i.  e.  call  the  constant  term  log  c,  instead  of  0,  we  have  y  =  log  (1  +  x2}  + 
logc  =  log  [c(l  +  #2)].  or  log  (c  +  ex?}. 


Ex.  22.  Integrate  dy  = 


SOLUTION.     —  g8  —    g      ^  _     x        x  ~      x    ^   jn  which  the  numerator 
2x-J  —  bx2  +  1  6       2x-  —  bo2  +  1 

/x"  —  2a! 
9  ••;  _  GX^  4-1       ^ 


112  THE  INTEGRAL   CALCULUS. 

Ex.  23.  Integrate  dy  =        —-.         V  = 


Ex.  24.  Integrate  dy  =  -= 
1  — 


.-**)  *]=tog- 


Ex.  25.  Integrate  ety  ==  --  ^—  .  2/  =  logc(a  -f 

Ex.  26.  Integrate  dy  —  —       ;—  .  y  =  log  -  -  -  7- 

(8a  —  3o^)* 

Ex.  27.  Which  of  the  following  can  be  integrated  by  the  method 

x  _  i 

used  in  the  last  6  examples  ;    viz.,  dy  =  —  —  -dx  •   dy  = 

ox~  —  Ax  -j-  1 


2x>  x  —  x>,  &r-dx  2a  —  ICte* 

___,»  ;  dy  =  —  - 

Ex.  28.  Integrate  dy  = 


SUG'S. 


x 


a*dx 
-   . 


1 
Ex.  29.  Integrate  dy  =  (6  —  &yx*dx. 

2.  I.  JUL  JJL 

Ex.  30.  Integrate  rfy  = - — dx. 

37 '' 


Ex.  31.  Integrate  dy  =  3  log2  #  —  . 


Sua's.     y  =  J~3  X  (logx)2  X  —  =  log3  *  +  ^»  since  —  =  d.  logx. 

dx 
Ex.  32.  Integrate  dy  =  2  log'  x—  .  y  =  $  log4  x  +  C. 

3C 

*  In  all  these  examples  c  represents  the  constant  of  integration. 


DEFINITIONS   AND   ELEMENTARY   FORMS.  113 

Ex.  33.  Integrate  dij  =  mlog"^— .  y  =  -  — -logn+l  x  -f  C. 

Ex.  34.  Integrate  dij  =  a2*  log  adx. 

SUG'S. — In  order  to  make  this  conform  to  (167 9  3),  we  should  have  d(%x)  =  2dx 
as  a  lactor.  Hence  we  write  y  =  )  a2xlog«d.£  =  J  sa^loga  •  2dx  =  ia2*  --f-  (7. 
[The  pupil  should  differentiate,  verify,  and  so  fully  consider  the  case  as  to  see  the 
reason  for  the  introduction  of  the  constant  factor.  ] 

Ex.  35.  Which  of  the  following  can  be  integrated  by  (167,  3)  ; 
dy  =  a?  log  a  Idx,  or  dy  =  3a*2  log  axdx? 

Ans.t  The  latter,  y  =  f  a**  +  G. 

X  X 

Ex.  36.  Integrate  dy  =  e"dx.  y  =  a?  +  G. 

Ex.  37.  Integrate  dy  =  3eTfe 
Ex.  38.  Integrate  cfy  =  6a3'<to. 

SUG'S.     y  =  I  ]a**dx  = fa3*  log  a  3dx  =  — asx  -4-  C. 

3  log  aj  3  log  a 

Ex.  39.  Integrate  dy  =  memdx.  y  =  ™en*  +  G. 


Ex.  40.  Integrate  dy  =cos  (2x)dx. 

SUG'S.  —  In  order  to  make  this  conform  to  (167  9  4),  we  should  have  2dx,  i.  e.  the 
differential  of  the  arc  2x,  instead  of  dx.  Hence  y  =  J'cos  2xdx  =  §  J"cos  2x  •  2dx 
—  ±  sin  2x  -f  C. 

Ex.  41.  Which  of  the  following  forms  can  be  integrated  by 
(167,  4)  ;  dy  —  cosa792c?j7,  or  dy  =  cos^xcfe? 

-4  /is.,  The  latter,  ?/  =  isin.r2  +  G. 

Ex.  42.  Integrate  cfy  =  sin3  x  cos  arrfa;. 

SUG'S.  y  =  J^sia'aoosasdas  =  J"l  X  (sin  a;)3  X  cosxdx  =  4sin-«x  +  C,  accord 
ing  to  (165  and  167,  4). 

Ex.  43.  Integrate  f/?/  =  sin  (3x)dx. 

y  —  —  £  cos  (3^)  -f  (7,  or  i  vers  (3a?)  +  (7. 

Sen.     4  vers  (3.r)  +  C  =  i  [1  —  cos  (3a?)]  +  ^  =  i  —  4  cos  (3a?)  -f  G, 

.',    C"==(7+^. 

Ex.  44.  Integrate  c??/  =  sin2  (2,r)  cos 


-f 
Ex.  45.  Integrate  rf?/  =  cos2  (3jl)  sin  (3.r)r/.r. 


114  THE  INTEGRAL  CALCULUS. 

Ex.  46.  Integrate  dy  =  sec2  x*xdx.  !/  = 

Ex.  47.  Integrate  dy  =  5  sec2  tf3  •  J7srfar.  y  =  ^  tan  a:3  +  (7. 

Ex.  48.  Integrate  dy  =  ^sec  (4#)  tan 


y  =  isec(4r)  +  (7. 

Ex.  49.  Integrate  dy  =  2  sin  (a  -f  3#)d.r. 

y  =  •  —  |  cos  (a  -\-  3«r)  -f  C. 

Ex.  50.  Integrate  dy  =  f  cosec2  v/2»  •  aT^dx. 

SUG'S.      y    =    J~f  cosec2  v/^x  •  x    dx    =    -  —  J*cosec2  v/^c  -  £\/2  •  a;    dr 

\/2 

o 

--  -  cot  v^x  4-  C*. 

v/a 

Ex.  51.  Integrate  dy  =  2  cosec  (no;)  •  cot  (nx)dx. 

o 
y  =  --  cosec  (n.r)  4-  (7. 

Ex.  52.  Integrate  dy  =  e*nxcosxdx.  y  =  esi11*  +  (7. 

Ex.  53.  Integrate  dy  =  —  eco8Xsmxdx.  y  =  ec('*x  -f  C. 


Ex.  54.  Integrate  dy  =  -  rTT-  y  =  2  tan  (^ar)  +  C. 

* 


Ex.  55.  Integrate  dy  =  -  —--  y  =  $coi  (3*2)  +  C. 


Ex.  56.  Integrate  dy  =  sin  (ax)dx. 

y  =  -  versin  (ax)  -{-  C,  or  —   -cos  (ax)  +  C". 

Ex.  57.  Integrate  ^y  =  —  cos  (£j;8  )#<&&. 

t/  =  covers  (-Ja;8)  +  (7,  or  —  sin  (£#»)  +  C'. 

Sen.—  In  the  last,  C'  =  C  -f  1.     In  the  56th  ^.u,  C"  =.  (7  +  -. 


Ex.  58.  Integrate  dy  = 


V/l  —  iU7^ 

SUG'S.  —  T/ie  /orm  of  </ie  denominator  suggests  at  once  that  this  may  be  the  dif 
ferential  of  some  arc  in  terms  of  its  sine.  Observing  that  the  numerator  is  the 
differential  of  the  square  root  of  4x2,  we  are  enabled  to  conclude  that  y  = 


C 

J  v/1 


-  =  sin-»  2o;  -f  C. 


v/1  - 


Ex.  59.  Integrate  dy  =  ——  .  y  =  i  sin~l  (x*)  +  (7. 

Vl  —  ar* 


DEFINITIONS   AND   ELEMENTARY   FOKMS.  115 

dx 


Ex.  GO.  Integrate  dy  = 

dx  dx  N/2 

SUG  S. 


A* 


— ' — .  •••  y=  I 

2N/1  — Sx*  J    v/2-9 


Sx.  61.  Integrate  cfy  = 


xdx  r       -xx         =  —    yxi;          ^ 

~5^   "J   v/2  v/i~=TS        J  2^f  v^5  v/i  -  ^ 


Ex.  62.  Integrate  ^y  =  -jfi-^  V  =  *  ^^  Y  +  a 

Ex.  63.  Integrate  <fy  =  ~=.  y  =  tsbrl$£)  +  Cl 


1      .  _!^8    .    ^ 
Ex.  64  Integrate  dy  =  ='  *  =  26  g        T  ' 


Ex.  65.  Integrate  ^y  =     -y  y  =  i  tan"1  ^  + 

Ex.  66.  Integrate  dy 

Site- 

Suo's.—  As  far  as  the  variable  is  concerned  this  conforms  to  the  differential  of 
an  arc  in  terms  of  its  versed  sine.  Thus,  the  numerator  =  dSj  Gx  ,  as  far  as  the 
variable  is  concerned  ;  and  x*  =  (x*)2,  which  is  the  relation  between  the  functions 
of  the  variable  in  the  denominator  of  the  form  referred  to.  Hence,  if  we  can  adjust 
the  constant  factors  to  this  form,  the  integral  will  be  apparent.  To  effect  the  latter, 
we  proceed  as  follows  : 


8.r  'dx 


_ -_  ^ 

\  2  .  dc3  —  (6x  )2  NJ  2  •  6x   —  (60;  )2 

in  which  6x*  being  regarded  as  the  variable,  the  expression  has  the  desired  form. 
»*)  +  C. 


116  THE   INTEGRAL  CALCULUS. 


Ex.  67.  Integrate  dy  =  —  -=  .          y  =         sec"1  -      +  0. 
xV  3x*  —  5  V5  * 


O  M — ^  rj  T 

Ex.  68.  Integrate  dy  = 


—  3 


MISCELLANEOUS  EXERCISES  UPON  THE  ELEMENTARY  FORMS. 

[NOTE.—  The  following  exercises  are  given  without  the  integrals,  as  it  is  of  first  importance  in 
the  Integral  Calculus,  that  the  pupil  be  able  to  discover  in  the  differential  the  probable  form  of 
the  integral.] 

Ex.  1.  Integrate  dy  = 


x  -f  cos  a; 


Ex.  2.  Integrate  dy  =  (2#3  -f-  x  l)dx. 

Ex.  3.  Integrate  dy  =  —     —  ;  also  dy  =  ~ 
1  -\-  x4  1  - 

xdx 
Ex.  4.  Integrate  dy  =  — 


v/1  —  ^2 

ra 4^  _^_  3 

Ex.  5.  Integrate  dy  =  -      -— -       —dx. 


IT  ff  7* 

Ex.  6.  Integrate  dy  =     "^7  -  »   also  dy  =  - 

4  4-  9# 2  4  +  9; 

Ex.  7.  Integrate  dy  =  (1  +  cos  x)dx. 
Ex.  8.  Integrate  dy  =  -     ^7=- — —• 

Ex.  9.  Integrate  dy  = 


v/a2  +  x* 

Zfl-r 

Ex.  10.  Integrate  dy  =  - 


v  5^4  —  2a;5 
Ex.  11.  Integrate  dy  =  cos5  x  sin  #<£r. 
Ex.  12.  Integrate  dy  =  tan2  #  sec2  .r  d#,  or  (tan2  a;  +  tan4ar)dar. 

Ex.  13.  Integrate  dy  =  (a  -   —  +  cx 
Ex.  14.  Integrate  dy  =  (1  +  #)(!  — 


RATIONAL  FRACTIONS.  117 

Ex.  15.  Integrate  dy  =  -    — — . 

Ex.  16.  Integrate  dy  =  — - — - — — . 

id  _j_  ox  \ 

Ex.  17.  Integrate  dy  =  -  . 


Ex.  18.  Integrate  dy  = 


Ex.  19.  Integrate  dy  =  * 


SOLUTION. 


x  + 
dx  4dx 


3  -f  1  +  4#  -f  4#a 


which  is  the  form  for  the  differential  of  an  arc  in  terms  of  the  tan- 


1  -4-  2x      2da; 
gent,  except  that  the  numerator  should  be  d.    '•  — —  =  — .    .  .  y—  .  g 

33 


Ex.  20.  Integrate  du  = 

12 


a;* 


SUCTION  II. 
Eational  Fractions, 

SEPARATION  INTO  PARTS  BY  INDETERMINATE  COEFFICIENTS. 

[NOTE.  —  The  body  of  what  is  called  the  Integral  Calculus  is  made  up  of  Special  Expedients  by 
means  of  which  differentials  of  various  forms  can  be  reduced  to  equivalent  known  or  elementary 
forms  A  few  of  the  more  important  and  fundamental  of  these  processes  are  given  in  this  and 
the  two  succeeding  sections.] 

172.  DEF.  —  A  Rational  Fraction,  as  the  term  is  used 
here,  is  a  fraction  in  which  the  variable  is  affected  with  none  but  pos 
itive,  integral  exponents.  The  general  form  is,  therefore, 

cxm~2  ----    -  -  Ix  +  k 


mxn  -p 


118  THE  INTEGRAL  CALCULUS. 

173.  Prop.  1. — If  the  highest  exponent  of  the  variable  in  the 
numerator  of  a  rational  fraction  is  greater  than  that  in  the  denominator, 
the  fraction  can  always  be  converted  into  an  equivalent  expression  con 
sisting  of  a  series  of  monomial  terms  with  or  without  a  rational  fraction 
(as  the  case  may  be),  in  which  fraction,  when  it  occurs,  the  highest  ex 
ponent  of  the  variable  in  the  numerator  shall  be  at  least  1  less  than  the 
highest  exponent  in  the  denominator. 

x*  —  ax*  4-  bx  +  c  (6  +  l)x  —  (a  —  c)  x*  —  a3 

+ 


a2  -}-  ax  -f-  «2  5  etc. 

174.  Prop.  2. —  Whenever  the  denominator  of  a  rational  fraction, 
as  — ~y,  whose  numerator  is  of  lower  dimensions  than  its  denominator, 

is  real  and  resolvable  into  n  REAL  and  UNEQUAL  factors  of  the  first  degree, 
the  fraction  can  be  decomposed  into  n  partial  fractions  of  the  form 

Adx          Bdx          Cdx  Ndx 

~r  — ; — i — r 


x-f-ax  +  bx-fc  x  -f-  n' 

and  these  fractions  integrated  separately. 

DE«.-Assume   M  _  _^_  +  _£-  +  _^_  .....  _JL.    Bringing 

^  ~-  - 


the  second  member  to  a  common  denominator,  each  numerator  will  be  multiplied 
by  7i  —  1  factors  of  the  form  x  -j-  m,  and  hence  will  be  of  the  (n  —  l)th  degree. 
Collecting  the  coefficients  of  a;"—1,  x"—  2,  etc.  in  the  numerator  of  the  second  mem 
ber,  it  will  take  the  form  Mxn~l  +  Nx»~*  -£  ---  Px  +  Qx»,  in  which  M,  N,  ---  P, 
and  Q  are  functions  of  A,  B,  C,  etc. 

Now/(«)  =  Mxn~l  +  NX*-*  -\  ----  Px  +  Qx°,  since  the  denominators  of  both 
members  are  equal.  Then  as  f(x)  is  not  above  the  (n  —  l)th  degree,  it  can  be 
treated  as  a  complete  polynomial  of  this  degree  ;  and  the  coefficients  of  the  like 
powers  of  x  being  equated  by  the  principle  of  indeterminate  coefficients,  there  will 
result  n  simple  equations  between  A,  B,  C,  etc.  ,  from  which  these  coefficients  can  be 
determined.  The  values  of  A,  £,  C,  etc.,  thus  determined,  being  substituted  in 
the  assumed  series  and  the  factor  dx  introduced,  the  decomposition  is  effected. 

Finally,  as  these  numerators  are  independent  of  the  variable,  we  have    /  - 
=  A  log  (x  -f-  a),  etc. 


.  Prop.  3. —  Whenever  the  denominator  of  a  rational  fraction, 

as  ,  whose  numerator  is  of  lower  dimensions  than  its  denominator. 

<P(*} 

is  real  and  resolvable  into  n  REAL  and  EQUAL  factors  of  the  first  degree, 
the  fraction  can  be  decomposed  into  n  partial  fractions  of  the  form 


BATTONAL  FRACTIONS.  119 

Adx  _  Bdx  _  Cdx  Ndx 

\  n     '       /  „,      i      „.  \  n — 1   ~T~     /  „ 


(x  +  a)"       (x  +  a)"-        (x  +  a)-2  x  +  a' 

and  these  fractions  integrated  separately. 


_j_  +  __.+___  _  To 

(x  +  a)n       (x  H-  a/'-1  ^(x  +  a/«-2  x  +  a 

bring  the  terms  of  the  second  member  to  a  common  denominator,  we  have  to 
multiply  B  by  x  -{-  a,  C  by  (x  -f-  a)2  .....  and  JV  by  (x  -f  a)"-1;  hence  the  nu 
merators,  when  added  together,  will  make  a  polynomial  of  the  (n  —  l)th  degree. 
Equating  this  numerator  with  /(#),  the  coefficients  of  the  corresponding  powers 
of  x  being  placed  equal  to  each  other  will,  as  in  the  preceding  demonstration,  give 
rise  to  n  simple  equations  between  A,  B,  C,  ----  -  N,  from  which  the  latter  can 
be  determined. 

Having  separated  the  fraction  into  the  partial  fractions  as  proposed,  it  remains 
to  be  shown  that  these  can  be  integrated.  As  the  numerators  are  constant,  and 
the  general  form  of  the  denominator  is  (x  -\-  a)n,  we  are  to  show  that  the  form 

-  —  -p-  —  -  can  always  be  integrated.  If  n  =  1,  /  -  -  =  log  (x  -f  a).  If  n  is 
anything  other  than  1,  /  -  —  --  —  -  =  f(x  -f  a)-"dx  =  —  —  -  (x  +  a)-*+S  or 

--  -  -  ;  --  -.     Hence  the  integration  can  always  be  effected. 
(n  —  l)(x  -f  a,)"-1 

Sen.  —  The  last  two  propositions  are  equally  true  whether  the  factors  of 
<p(x)  are  real  or  imaginary  ;  but  as,  in  the  latter  case  the  integration  by 
those  methods  would  give  logarithms  of  imaginary  numbers,  the  method 
given  in  the  next  proposition  is  preferable.  We  will  still  farther  premise 
that  as  <p(x)  is  real,  if  it  contains  imaginary  factors,  they  must  enter  by  pairs 
of  the  form  x  ±  a  +  bV  —  1  and  x  rb  a  —  b\/  —  1,*  for  only  thus  can  a  real 
product  arise  from  imaginary  factors.  If  therefore,  there  are  imaginary 
factors  in  <p(x),  we  shall  have 

<p(x)=if>(x)  •  {(x±a  +  b^/^i)'X(x±a  —  bV—i}}*=ii>(x}-  {  (a;  ±  a)*  +&»}", 
in  which  if>(x)  represents  the  product  of  the  real  factors. 


170.  I*rop.  4=.  —  Whenever  the  denominator  of  a  rational  fraction^ 

as     '  ,   —  .  whose  numerator  is  of  lower  dimensions  than  its  denominator^ 

<p(x) 

is  real  and  resolvable  into  n  real  and  equal  QUADRATIC  factors,  the  fraction 
can  be  decomposed  into  n  partial  fractions  of  the  form 

B)dx  (Cx  +  P)dx^  ,       (Ex-f  F)dx  (Mx+N)dx  ' 


and  these  fractions  integrated  separately. 

DEM.  —  [The  first  part  of  the  demonstration,  showing  that  the  fraction  can  be 
*  Called  conjugate  imaginary  factors. 


120  THE  INTEGRAL  CALCULUS. 

separated  into  this  form,  is  identical  with  that  of  the  last  proposition,  and  the 
student  can  supply  it.] 

Having  separated  the  fraction  into  partial  fractions  as  proposed,  it  remains  to 
be  shown  that  these  partial  fractions  can  be  integrated.     The  general  form  is 

(Ax  +  B)dx 
f/x  .+.  ayz    i    frqn'  m  wlnch  n  ls  an  integer.     To  reduce  this  to  known  forms  put 

x  ±  a  =  z,  whence  x  —  z  =F  a,  dx  =  dz,  and  (x  ±  a)2  =  z2.     Substituting  these 
values,  we  have 

y*  (Ax  +  B}dx       _    r(Az  g=  Aa  +  B]dz  _    /•     ^4zdz  /»(J? 

[(z  zb  a)2  +  6*]-  ~J  (Z2  _|_  i*)~      -J  (z*  _j_  b*-*  +J     (z* 

fA(z*  +  &*)~"zdz  +   A-j~~-,  in  which  ^l'=  .S  =F  .4a. 


By  (105)  we  have   A(z*  +  &2)-"zdz  =  —  - 


*  In  a  subsequent    article   (192,  formula  £T5}»)   it  will  be  shown    that    the 
I  — a  -L.  /2~^»  ma^  ^e  ma(^e  ^°  depend  upon    /  — 2    \    i..2  „_ i>  wnich  in  turn  may  be 

made  to  depend  upon   /     ..    .  '\    n_a>  thus,  in  the  end  giving  either   /  - 


»  SCH.  —  In  case  the  factors  of  the  denominator  are  not  readily  seen, 
put  it  equal  to  0,  and  solve  the  equation  for  the  variable.  According  to  the 
theory  of  the  composition  of  equations,  as  developed  in  Higher  Algebra, 
the  variable  minus  each  of  the  several  roots  in  turn  will  be  the  factors. 


Ex.1.  Integrate  dy  = 


x*  -f  &c*  +  llx  -f  6 


SOLUTION.— Putting  aP  +  6#2  +  llx  +  6  ==  0,  we  find  x  =  —  1,  —  2,  and  —  3.+ 
.  x3  +  6#2  +  llx  +  G  =  (x  +  l)(x  +  2)(x  +  3),  and  we  assume 
x2  +  1  A B_ C__ 

~       t       O        I*    ™       I       •) 


.j  _|_  tja;2  +  llx  -f  6        a;  +  1    '    «  -f-  2 
g  +  3)  J?(y  4.  l)(a;  +  3) 


(x  +  l)(a;  +  2X»  +  3)         (.«  +  IX*  +  2;^as  +  3)    '    (x  +  lj^  +  2)(x  +  3) 
Ax*  +  5Ar  +  6^1  +  Jg.r2  +  4^.r  +  35  +  C&  -\-3Cx  +  26' 
x'J  +  Gxa  +  llx  +  ti 

Whence  x2  +  1  =  (A  +  J5  +  (7)x2  +  (5J.  +  45  +  3C')x  +  6^1  +  35  +  2C1 

These  members  being  identical, 
A  +  5 +  (7=1    0)5     5^1+45  +  36f=0    (2);     and  64  +  35  +  2(7=  1    (3). 

From  (1),  (2),  and  (3)  we  find  A  =  1,  B  =  —  5,  and  C=  5. 

Hence  we  have 

*  This  reduction  might  be  exhibited  here,  but  as  the  formula  referred  to  is  better  for  practical 
purposes,  it  is  thought  best  to  give  the  process  but  once, 
t  COMPLETE  SCHOOL  ALGEBRA,  Tart  II.,  (111). 


RATIONAL  FRACTIONS.  121 

l}dx  /•  dx          .    r  dx 


Ex.  2.  Integrate  cfy  = 
* 


-  log  e  = 
adx 


/adx       _  1   r  dx          1   r  dx      _          l&(x  —  a] 
x*  —  a?~  ~2J  ~x~^a  ~~  2J  lT+~a  ~      g\|     x  +  a  ~ 


(3#  —  5}dx 
Ex.3.  Integrate  rfy  =  L__J_. 


=  £log  (a?--  4)—  |log(^—  2)+logC—  log 


Ex.  4.  Intecrate  dy  = 

5* 


—. 
bx 


SUG'S.     •  -  —  —  =  —  -j  --  r  gives  a  =  y!6,  and  0  =  A  4-  B  ;  whence  ^4  =  -, 
x*-\-bx      x*      x  -j-  6  6 


b'   ' 

Ex.  5.  Integrate  dy  =  —  '     . 

y  =  i.  log  x  +  -|  log  (2  +  x)  -f  log  (2  —  x)  -f  (7. 

(  i\f  ) 

Ex.  6.  Integrate  dy  =  •izfli  y  =  log  ^  -^~ 

/i 
Ex.  7.  Integrate  dy  —  - 


x,  +  x  _ 


2  _  3^.2 
Ex.  8.  Integrate  dy  =  -  --  r-zrdx. 

{x  -j-  ^;B 

2_3x2  ^1  B  C         A  +  B(x  +2)+  C(x+  2) 

SuG's.-Assnme 


whence2  —  3x2  =  A  +  Bx  +  2B  +  Oc2  +  4Gc  +  4a     .'.  ^i+2B-f-4(7^2  (1); 
=  0  (2)  ;  (7=:—3  (3)  ;  from  which  .B  =  12,  A  =  —  10,  and  C=  —3. 

dx 


1  2 

—  - 


*  This  factor  is  equivalent  to  x  -f  0,  BO  that  a;2  _|-  fca;  =  (x  -f  0)(x  -f  6). 


122  THE  INTEGRAL  CALCULUS. 

3^,  _  1 

Ex.  9.  Integrate  dy  =  -  —  '•  —  ^-dx. 

(x  —  d)2 

y  =  -  ^13  +  3log  (*-  3)  +  C-. 

xydx 

Ex.  10.  Integrate  dy  =  -  -  —7  -  -. 
(x  —  a)*(x  +  a) 

Suo.  —  When  some  equal  and  some  unequal  factors  occur  in  the  denominator, 

x2 

the  assumed  forms  must  be  combined.      Thus  -  -  -  •  —  -  must  be  put 

(x  —  a)\x  +  a) 

=  —  -  --  L-JL-J  --  —  .     A  =  ^a,  JB  =  |,  and  C=l    2/=  —  «  —  -  -  -f 
(a;  _  a)2  ~  x  _  a  ~  x  -f  a  %(x  —  a)  T 


|log(x  —  d)  +  l\og(x  +  a)  +  C. 
Ex.  11.  Integrate  dy  = 


SUQ.     a;3  —  6x2  -f  9x  =  &(x2  —  6x  +  9)  =  x(x  —  3)2. 

y  =  i  log  ar  +  ^  log  (a:  —  3)'  +  i  log  c  =  log  [cx(x  — 

dx 


Ex.  12.  Integrate  rfy  = 


Sra-Assume  ——^  =  -  +  -.  +          -  +  -  whence 

=  -2\,  J  =  —  T|F,  C  =  -h,  and  D  =  Tf ,. 


Ex.  13.  Integrate  dy  =  ^  ^  2)2  **• 

SUG. — The    simple    binomial    factors   of  the    denominator   being    imaginary, 
(x  +  x/=2)»(a  -  VCT2)2,  we  assume  ^±p^  =  ^^  +  §^|  J  whence 

4  =  —  1,  J?  =  —  1,  C=  1,  and  D  =  0. 

/,r3_|_a;_i              /•_£_!               /»aate           /*  xda;         /• 
—  '     do:==    / dx  -4-    I =  / \  x(&  4-  2)-2daj 
^  _|_  2)2  *        J(a;2  +  2)2 u*^Jx*  +  2      J*+*     J 

—  J —  =  \  loe(x2  4-  2)  -I /" — — — .     The  last  term  can 

J   (x*  +  2)2  r2(«s  +  2)      ^/   (x^  +  2)^ 

be  integrated  by  formula  2J5>  (192,  Ex.  10). 
Ex.  14.  Integrate  dy  =  -• 


—  x*  +  2x  —  2 
Suo's.—  The  simple  factors  are  x  -|-  v/— 2,  a;  —  >/— 2,  and  JB  —  1 ;  and  the 


KATIONAL  FRACTIONS.  123 

t 

form  of  the  partial  fractions  is  :~^y  ,  ^  ^     ^  ==  —  i,  -B  =  —  i,  and  (7  =  i. 


-to-         +  (,, 


Ex.  15.  Integrate  dy  = 


Ex.  16.  Integrate  rfy  =  ^4   ,   ^  _  2' 

«2  yl  5       ,    Ge  +  D 

SUG'S.  -Assume  ^^^  =  ^p  +  —  ~,  +  —^  I  -^nce  A  - 

=  i(7=0,andD^|.    y  =  log  (j)*  +  ~  tan"         +  G 


Ex.  17.  Integrate  dy  = 
Ex.  18.  Integrate  c?y  =  - 


Ex.  19.  Integrate  dy  =  — 

Ex.  20.  Integrate  dy  = 

y  =  *£  +  -flog  (x  +  2)  +  I  log  (x  —  2)  +  a 

jf7£.  Sen.— It  will  be  observed  that  the  foregoing  processes  of  separa 
ting  rational  fractions  into  partial  fractions  make  .their  integration  depend 
on  one  or  more  of  the  following  forms  : 

c  m   r  dv   f dx  -  r xdx   c  xdx    c  dx 

All  of  these  forms  except  the  last  are  integrable  by  the  elementary  pro 
cesses.     The  integration  of  the  last  is  effected  by  formula  *&  (192}. 


124  THE  INTEGRAL  CALCULUS. 

• 

SECTION  III. 
Eationalization, 

170 •  When  polynomial  radicals  occur  in  a  differential  which  we 
desire  to  integrate,  it  is  sometimes  possible  and  expedient  to  rational 
ize  the  expression  by  the  substitution  of  a  new  variable  which  is  some 
definite  function  of  the  variable  in  the  given  differential.  A  few  of 
the  more  important  cases  are  given  in  this  section. 

BINOMIAL    DIFFERENTIALS. 

180.  Prop.  1. — Every  binomial  differential  can  be  reduced  to  the 
form,  xm(a  -j-  bxn)pdx,  in  which  m  and  n  are  integral,  and  n  positive. 

DEM. — 1st.  If  x  occurs  in  both  terms  of  the  binomial,  and  the  form  is 
xr(ax?  -j-  bxt)fdx,  we  can  remove  from  the  parenthesis  the  factor  x*,  or  x1,  which 
has  the  less  exponent.  Thus  suppose  s<^t,  we  can  write  xr(ax'  -j-  bx1')  P  dx  = 

/  7*'  \P 

xr  '  XPS(  a  -j-  b—  J  dx  =  or+^a  -f-  b&~')*dx.  In  this  form  t  —  s  is  positive,  since 
t^>  s,  but  it  may  be  fractional,  r  -j-  ps  may  be  either  positive  or  negative,  integral 

1*  6 

or  fractional.      Now  let  r  -\- ps  =  ±  ~,  and  t  —  s  =  -j-  -  ;    whence  we    have 

x  \a-\-bx    f)Pdx. 

±-  +'- 

2nd.  In  the  latter  form  put  x  =  zllt ;   whence  x    h  =  z±c/,  x  f  =  z+eh,  and  dx  — 

±-  +- 

hfzhJ—ldz.        Substituting     these     values,     we     have     x    h(a    -f-    b,x   ')t>dx    = 

z±c/(a  -j-  bz+eh)p}ifzhs~}dz  =  hfz±cS+''f—}(a  -f-  bz+eh)vdz,  in  which  the  exponents 
of  z  are  integral,  since  c,  e,  /,  and  h  are  integers,  and  eh  is  positive.  Therefore 
putting  it  tf-\-hf — 1  =  m»  and  eh  =  n,  we  have  hfzm(a-\-  bz")vdz.  Q.  E.  D. 


181.  Prop.  2.—  A  binomial  differential  of  the  form  xm  (a,  +  bxn)«dx 
(  any  or  all  the  exponents  being  fractions)  may  be  rendered  rational  by 

putting  a  +  bxn  =  zq,  when  —       -  is  integral 

DEM.—  Putting     a  -f  bx»  =  z*,  (1) 

we  have  .     (a  -f-  b  x")5  =  ZP,  (2) 

Differentiating  (1),  nb  xn~ldx  =  qzi-^dz.  (3) 


Also  from  (1),  &*-"+'  =  C* 


+t 


Multiplying  (2),  (3),  and  (4),  nbxm(a  -f-  bx»)*>dx  =  qz*«+«-*—  dz, 

p  q__ 

or 


RATIONALIZATION.  125 

Now  by  hypothesis  p  -J-  q  —  1  is  integral ;  hence,  if  —     -  is  integral ,  the  expression 
is  rational.     Q.  E.  i>. 


182.  Prop.  3.—  A  binomial  differential  </^e/ormxm(a+bxn);»dx 
(any  or  all  the  exponents  being  fractions]  may  be  rendered  rational  by 

putting  a  +  bxn  =  zqxn,  when  -  ---  h  -  is  integral. 

DEM.—  Putting     a  -f  bxn  =  z'z",  (1) 

we  have  xn  =  -  -,  (2) 


_  _ 

-  T  =  a"(2«  —  b)  ",  (3) 


and  xm  —  a  "(z9  —  &)". 

Multiplying  (2)  by  &  and  adding  a,  we  have 


_ 

whence  (a  -f-  &x")«  =  a«(z«  —  &)    «ZP.  (6) 


Differentiating  (3),  dx  =  —  ^a»(2«  —  fo)    "  "z«-'dz.  (7) 

Multiplying  together  (4),  (6),  and  (7)  and  putting  A  for  the  constant  factor,  there 
results 

E  fm  +  1    |    P   |    !\ 

=  ^l(z«  —  6)    ^    "       « 


Now  by  hypothesis  p  -f  5-  —  1  is  integral  ;    hence,  if  -      --  (-  -  is  integral,  the 
expression  is  rational.     Q.  E.  D. 

183.  Sen.  —  Although  the  rationalization  can  always  be  effected  as  stated 
in  the  last  two  propositions,  it  does  not  always  facilitate  the  integration. 

When  in  the  former  case  -  —  1  is  a  positive  integer  or  0,  or  in  the 

latter  ^—  -  --  1-  -  -j-  I  is  a  negative  integer  or  0,  the  binomial  zq  —  b  will 
n  q 

have  a  positive  integral  exponent  and  can  be  expanded  into  a  series  of  a 
finite  number  of  terms,  or  a  0  exponent  and  will  be  equal  to  1.  Hence  in 
any  such  case  the  rationalization  will  lead  directly  to  the  integration.  But 

if  !!  _L  --  1  is  a  nearative  integer  in  the  former,  or  —  3-  —  4-  —  4-  1  is  a 
n  n  q 

positive  integer  in  the  latter,  the  exponent  of  sfl  —  b  will  be  negative,  and 
the  rationalization  will  not  generally  lead  to  the  integration  ;  and  in  fact  it 
is  not  usually  expedient  to  rationalize  in  such  cases. 


126  THE   INTEGRAL   CALCULUS. 

184.  COR. — Every  differential  of  the  form,  dy=  Axm(a-fbx)pdx  can 
be  rationalized  and  integrated  when  either  m  or  p  is  a  positive  integer. 

DEM. — If  p  is  a  positive  integer,  a-^-bx  can  be  expanded  into  a  series  of  a  finite 
number  of  terms,  which  multiplied  by  xmdx  will  give  a  series  of  monomials  ;  and 
an  algebraic  monomial  can  always  be  integrated  by  (165  or 


If  p  is  fractional  or  negative  and  m  integral  and  positive,  —         —  1  =  — — 1, 

will  be  a  positive  integer  or  0,  and  Prop.  2,  will  effect  a  rationalization  which  will 
lead  directly  to  the  integral. 

Ex.  1.  Integrate  dy  —  #5(2  -f  %xz}*dx. 

SUG'S. — Since  m  =  5,  and  n  =  2,   — ^—  —  1  =  2,  a  positive  integer,   and 

Prop.  2,  will  lead  to  the  integration. 

To  rationalize,  put      2  -f  3x*  =  z«;  (1) 

whence  (2 -f  3x')  *  =  z,  (2) 

Differentiating  (1)       xdx  =  $zdz,  (3) 

Also  from  (1)  x4  =  ( — - — j  .  (4) 

Multiplying  (2),  (3),  and  (4)  together 

dy  =  #5(2  -f-  3x2)  dx  =  ^V(z2  —  2)2z2dz  =  £r(zedz  —  4z4dz  -f-  4z2<2z). 
.  • .  y  =  -^fj(zcdz  —  4z4dz  -f-  4z2dz)  =  j^f —  -j j  -|-  Q ;  or,  restoring  the 

value  of  z,   y  —  -^  -j  — ^J~ <f~~     ^ 3~~~  ~  f   +  ^      t™8 

result  may  be  expanded  and  reduced,  if  desired.  ] 

Ex.  2.  Integrate  dy  =  #3(a  -f  bx^dx. 

Ex.  3.  Integrate  dy  =  x*(a  —  x*)~2dx. 
Ex.  4.  Integrate  dy  =  ar5(a  +  x)2dx. 

Ex.  6.  Integrate  dy  = -. 

SUG'S.     y  =  f '• j  =  J  ar~4(l  -}-  a2)    dx     Here  m  =  —  4,  n  =  2,  and 


=  /*- 

J  x"(i  + 


p  =  —  £  ;  whence  -        -  -f  "  +  1  =  _JL± ^  +  1  =  —  1,  and  Prop.  3, 

•will  lead  to  the  integration. 


RATIONALIZATION.  127 


Putting   1  +  x*  —  z»x2  (1)  ;    we  have  &  = (2)  ;    x  =  7   (3)  ; 

Z     1  -  .  2 


and  differentiating  (3),  dx  =  —  (z2  —  l)~*z  dz  (7). 
Multiplying  together  (4),  (6),  and  (7),  there  results 


J  v  3#J 

Ex.  6.  Integrate  dy  = • -. 


o  /»/>• 

Ex.  7.  Integrate  %  =  a(l  +  x~)~^dx.  y  =  -          -—+  C. 

Ex.  8.  Integrate  dy  =  0^(1  —  2tf*; 


IRRATIONAL    FRACTIONS. 

.  Prop.  1.  —  When  a  fraction  contains  none  but  monomial 
surds,  it  can  be  rationalized  by  substituting  a  new  variable  with  an  expo 
nent  which  is  a  common  multiple  of  all  the  denominators  of  the  fractional 
indices  in  the  given  expression. 

DEM.  —  The  general  form  of  such  a  fraction  is 

m  p 

axn  4-  bxq  -4-  etc.  . 
—7  --  T  ---  dx' 
ax"       Vxu       etc. 


In  this  put  x  =  z"*™'  etc-  ;  whence  »"   ==  z"1''"'  etc-,  xq  =  znp*u>  et<S  x"  =  2^^'  etc-, 

t 

att  __  znqrt,  etc^  an(j  ^  _;  (nqsu,  etc.)  2n«8"-  ctc-  ~'dz.     These  values  substituted  in  the 
given  fraction,  evidently  render  it  rational. 

186.  COR.  —  This  method  is  equally  applicable  wlien  the  fraction  in- 

m 

volves  no  surd  except  one  of  the  form  (a  +  bx)  »,  by  treating  a  -f  bx  as 
the  variable. 


18V.  Prop.  2.  —  When  a  fraction  contains  no  surd  but  one  of  the 

form  v/a  -{-  bx  =b  x2,  it  can  be  rationalized  by  putting  \/a4-bx-fx2  ==  z  —  x 


128  '  THE   INTEGRAL   CALCULUS. 

when  x2  is  +  ;  and  when  x2  is  — ,  \/a  +  bx  —  x2  =  v  (x  —  &)(P  —  x) 
=  (x  —  of)z,  in  which  a  and  ft  are  the  roots  of  the  equation 
a  +  bx  —  x2  =  0. 

DEM.— 1st.    When  xn-  is  -f.     Putting  Va  +  bx  -f  x*  =  z  —  x,  a  +  bx  -f-  x*  = 
z*  —  2zx  -far3 ;  whence  #  ==  j^T^'  ^  =        ->T    ^        2>  and  ^"-MM-"*5  = 


z  —  yTTTT;  ~  —  uz  -4-  b     '     Hence  as  £C'  Vo^Fte  -f-  a;2,  and  dx  are  expressed  in 
rational  terms  of  z,  the  transformed  fraction  will  be  rational. 


2nd.    When  xn-  is  —  .     Assuming  \/a-\-bx  —  X*  =  v"(x  —  a)(fi—x)  =  (x—  a)z, 
and  squaring  we  have  (x  —  a)(/3  —  x}  =  (x  —  a)?z2,  or  fi—  x  =  (x  —  a)z^  ;  whence 


,  and 


—  —  —-  .     Hence  as  x,  V  a  -\-ltx  —  x2,  and  dx  are  expressed  in  rational  terms  of 


the  transformed  fraction  will  be  rational.     Q.  E.  D. 

1  2 

2r2  _  3.^5' 
Ex.  1.  Integrate  dy  =  —-  -  —  —  dx. 


SUG'S.—  Put  x  =  z*  ;  whence  dy  =  -^z7  dz  —  V1*8  dz.     y  =  iW  — 


Ex.  2.  Integrate  cfy  = 


Ex.  3.  Integrate  dy  = 


G       T^      4r2  1 

+  -7  +  -'^-4-^^ 

^j  O 


SUG'S.  —Putting  1  -f  x  =  z2,  %  =  —  -—  -  =  —^7—=  5   whence  y  =  2  tan-'(l  -f-  x) 
+  G 

Ex.  4.  Integrate  dy  =  —        —  -. 
(1  +  4ar^ 


Ex.  5.  Integrate  dy  =  —  -:  --  .         y  =  log  '—       -  -f  C. 

XVI  +  X  V  1   +  J7  +  1 

SUG.  —Ex's  4  and  5  can  be  performed  by  (184)  or  (181).     In  fact  these  methods 
are  essentially  identical  when  there  is  but  one  surd  of  the  form  (a  -}-  ?>.t)"  . 


EATIONALIZATION.  129 


Ex.  6.   Integrate  dy  =  — . 

xv  1  4-  x  -f  x* 


SOLUTION.  —Put    \/l  -f-  x  -f  A-*   =  2  —  x  ;    whence  z  —  a;  +  v/1  -f-  a;  -f-  a*, 


—  1  2(z*  -f-  z 


--  - 

35  =  air  i'    =  -larFi—  •  and  vi  +  *  +  *•  =  -ir+Tn  Substitut- 

ing  these  values 

=  /•      **        .  /-2^+s+p    fc+i    _2MLi,jz    r  a** 

./  .^'iT^T^    J    ^  +  ^  .    2^  -  1     *+z+l      J  *-^~ 


B.1-14-V14-B-L-* 

[ i^r:i=rzir  -f-  G  — 

l{B  +  1-t-\/l-4-a:4-x* 


log —  +  C. 

2  -f  x  +  2v/l  +  a;  +  ar8 

Ex.  7.  Integrate  dy  = 


x  —  1 


Ex.  8.  Integrate  dy=—  '—-. 

SUG'S.  — Putting  V^x-^-x'2  —  z  —  x,  there  results,  in  the  usual  way  dy  = 


^    = 


j  +C=  log  (a;  +  1  -f>/2x  +  a»)  --  — 

*  T 


v/2ar  -f- 


Ex.  9.  Integrate  dy  = 


SOLUTION.— Put  v/2  —  x  —  x2  =  v/(a?  +  2)(1  —  x)  ==  (x  +  2)z  ;  whence  x  = 

1  —  2z2  62^2  , Sz 

_   rinf  —  —  and  v2  —  x  —  x2  = •     v 

z*  +  l 


Ex.  10.  Integrate  cf?/  = 


N/l 


57 


Suo's.—  From  1  -f-  #  —  x2  —  0,  we  learn  that  the  factors  are  x  —  (£  -}-  iv/5)  and 
—-  |v/5)  —  «.  As  these  roots  are  so  cumbrous  it  will  be  economy  to  take  x  —  a  and 
—  x  as  the  factors,  as  in  the  general  demonstration.  The  differential  in  terms  of 


130  THE  INTEGRAL  CALCULUS. 

SECTION   IV. 

Integration  by  Parts, 

188.   The    Formula  for   Integration   by  farts  is 

j"u  dv  =  uv  —  fv  du. 

This  formula  is  deduced  directly  from  the  differential  of  a  product.  Thus  d(uv) 
=  v  du  -f-  u  dv  ;  whence  uv  =  j^vdu  -{-  J*udv,  and  J^udv  =  uv  —  fvdu. 

FORMULA    OF   REDUCTION    «&,    JB,    <g,    AND    J&. 

.M.9.  JProb.  —  ^o  produce  a  formula  for  reducing  the  exponent  of 
x  without  the  parenthesis  by  the  exponent  of  x  within,  in  the  form 
dy  =  xm(a  +  bxn)pdx  ;  i.  e.  to  make  the  integration  of  this  form  depend 
upon  the  form  /x'n-u(a  +  bx")pdx. 

SOLUTION.  —  The  solution  of  this  problem  is  effected  by  applying  the  formula  for 
integration  by  parts  to  the  form  dy  =  xm(a  -J-  bx")Pdx.  To  make  the  application, 
put  dv  =  (a  -f-  bxn)Pxn-ldx,  whence  u  =  £"<-"+  >.  , 

Integrating  the  former  and  differentiating  thVlatter,  we  have  v  =  -  —  --  —  -  -  , 

nb(p  +  1)    v 

and  du  =  (in  —  n  -f-  l)xm~"dx. 

Substituting  in  the  formula  fu  dv  =  uv  —  fv  du,  we  hare  y  =  fxm(a  -j-  lxn)?dx 

ym-n+l(         I     lj.xv,p+l  m  -  n  _|_  1    r 

=  -  ,     T  t  --  —,  -  rrr  '  xm~n(a  +  ^n)p+}dx.*    (i). 

nbj)  -f  I)  nb^p-\-  I,  " 

Now  fxm~n(a  -f-  l)x»y+i<lx  =  fxm~n(a  +  bx«)pdx  X   (a  +  6x»)  = 

ftJV'-"(«  +  bxn]pdx  +  bfxm(a 
Introducing  this  value  of  fxm^l(a  -J-  bxny+}dx  in   (1),  we  have 


(a  i 


Transposing  the  last  term,  we  have    1  1  -J  --  •     ?     ,          j-  fx»'(a  -f  frx")7^ 


o(m  —  n  +  D.^TO~n(«  4- 


*  This  operation  diminishes  m  by  n,  but  it  also  increases  p  by  1.  The  latter  may  be  a  disad 
vantage.  It  will  evidently  be  more  likely  to  prove  advantageous  if1  we  can  diminish  or  increase 
either  m  or  p  without  affecting  the  other. 


INTEGRATION   BY   PARTS.  131 

100,  Prob.  —  To  produce  a  formula  for  increasing  the  exponent  of 
x  without  the  parenthesis  by  the  exponent  of  x  loithin,  in  the  form 
dyj~xm(a  -|-  bxn)pdx  ;  i.  e.  to  make  the  integration  of  this  form  depend 
upon  the  form  /xm+n(a  +  bxn)pdx. 


SOLUTION.  —  Clearing  (,&)  of  fractions,  transposing  the  first  member  and  the  last 
term  of  the  second  member,  and  dividing  by  a(m  —  n  -f-  1)>  we  have 

xm-"+1(«  _j_  &o7»)p+i  —  l>(np  4-  ra  4-  1)  fxm(a  -|-  lxnYdx 

I  xm~  n(a  4-  bx*}Pdx  =  --  -  r-TTr  -  :  ---  • 

J  a(m  —  n  -j-  1) 

Putting  m  —  n  =  m',  whence  m  =  m   -\-  n,  this  becomes 

r  xm'+i(a+bxn*)P+l  —  l>(np-\-m  -\-n-\-l)  fxm'+n(n-\-bxn']fdx 

y  =    xm'(a-\-ljxn}i>dx  =  .  -  -  •  --  ,    ,    --  -  —  --  5 
y      J       v  a(m  4~  1) 

or,  dropping  the  accents, 

y  =    xm(a 
y      ^       v 


-f-  1) 


101.  IProb.  —  To  produce  a  formula  for  diminishing  the  exponent 
of  the  parenthesis  by  1,  in  the  form  dy  =  xm(a  +  bxn)pdx  ;  i.  e.  to  make 
the  integration  of  this  form  depend  upon  the  form  /xm(a  -f  bxu)p~1dx. 

SOLUTION.  —  We  may  write 
y  —  JV-(a  +  bxn)Pdx  =  /xm(a  +  &x»)p-'dx  X  (a  +  bxn)  = 

afxm(a  4-  ftic")*-'^^  +  &/.'c™+n(a  +  5x»)^-1da;.   (1). 
By  applying  formula  (<.&)  to  the  last  integral,  it  becomes 

xm+'(a  4-  6x«>  —  rt'm  -f  1)  fxm(a  +  te")^-1^ 

'  •    xm+n(a  4-  Zw^y-'dx  =  --  .  -  :  -  r^i  ---  '• 
-;  b^np  4-  in  4~  1) 

Substituting  thin  in  (1)  it  becomes  y  =fxm(a-}-bxn)i>dx  =  a 
xm+i(a  _j_  6a;«)p  —  a(m  -|_  l)"a;TO(a  4-  &xn)p- 


wp  4"  m  +  i 

or,  uniting  terms, 

xm+l(a  4-  bxn]t>  4-  *7wp  fxm(a  4-  y 
-  ——->-— 


102.  IProb*  —  To  produce  a  formula  for  increasing  the  exponent  of 
the  parenthesis  by  1,  in  the  form  dy  =  xm(a  4-  bxn)pdx  ;  i.  e.  to  make  the 
integration  of  this  form  depend  upon  /x™(a  +  bxn)p+1dx. 


SOLUTION.  _  Transposing  and  reducing  (<g)  in  the  same  manner  as  we  did 
in  producing  (SS),  we  have 

a;w+i(a  4-  &xny  —  (np  +  m  -f  1)  f  xm(a  4-  ?;x")^x 


Putting  p  —  l=p',  whence  p=p'  4-1,  this  becomes 


132  THE  INTEGKAL  CALCULUS. 


- 
or,  dropping  the  accents, 


Sen.  —  Binomial  differentials  of  this  form,  or  such  as  may  be  readily  re 
duced  to  it,  are  of  such  frequent  occurrence,  and  the  formulae  g&,  2B,  <J£, 
and  SJ^,  called  Formulae  of  Reduction,  are  so  frequently  efficient  in  reducing 
them  to  known  forms,  that  these  formulas  should  be  carefully  memorized. 

Ex.  1.  Integrate  dy  =  -  --  -. 

•   (a2  —  x*y 

SOLUTION,    y  =  I  —  —  -  —  -    —  fx\az  —  xz)    dx,  a  form  which  corresponds 

J    a?  —  & 


to  fxm(a  -{-  bxn}Pdx,  by  considering  m  =  2,  n  =  2,  a  =  a2,  6  =  —  1,  and  p  =  —  £. 
We  now  observe  that  if  the  exponent  of  x  outside  the  parenthesis,  or  in  the  nu 
merator  in  the  given  form,  were  0,  so  that  x°  =  1  ,  the  integral  would  be  a  cir 
cular  function.  Now  formula  (s&)  will  so  reduce  this  exponent  ;  hence  we  apply 


it,  and  have  y  =  I = 

-  a?2)       '  —  <r2(2  —  24- 


-l[2(-i) +  2-1-1] 


Ex.  2.  Integrate  dy  =  - 

y  —  —  J#'(l  —  #s)2  —  §(i  —  a;j)i  4. 

Ex.  3.  Integrate  dy  =  - 

(2  4-  x*)* 

y  =  &(2  4-  x*)~^  4-  4(2  4-  &)~^  4-  (7=     ^  +  4.  4-  0. 

(2  4-  a;*)* 


INTEGRATION  BY  PARTS.  133 

Ex.  4.  Integrate  dy  =  • 


— Apply  gfikj  twice  in  succession,  and  we  have 


Ex.  5.  Integrate  dy  = 


1-5         1  •  3    5  \   /- —      13-5. 


dx 
Ex.  6.  Integrate  dy  =  - 


Suo.-Apply  Jg  twice,    y  =  —  ^'        V  —  **)*  + 


dx 
Ex.  7.  Integrate  dy  — 


=  /"  -  ~  -  T  =  fx~3(~  a* 
^  a;3ic2  —  a22 


SOLUTION,         =       -  ~  -  T  =     x~3(~  a*  + 


)    "^     Now  b^  increasing 

the  exponent  of  a;  without  the  parenthesis  by  that  within  the  form  becomes  known. 
Hence  we  apply  (53).  In  this  case  m  =  —  3,  n  =  2,  a  =  —  a2,  6  =  1,  and 
p  =  —  £.  Hence  substituting  in  the  formula 


y=  C  -  ^-— 
J  a3(a;2-a2) 


—  «2)2 


Ex.  8.  Integrate  <fy  =  (a2  — 
SUG'S.— Applying  &,  we  have 


r  ij 

Applying  <g  to  J  (a2  —  x*}  ax,  we  have 


134  THE  INTEGRAL  CALCULUS. 


Ex.  9.  Integrate  dy  —  (I  —  x*y>dx. 

SUG.—  Apply  <g  twice,     y  =  ^x(l  —  x*)*  -f  fx(l  —  #3)*  -f  f  sin-i  x  +  C. 


Ex.  10.  Integrate  dy  =  - — — . 

(a2  +  a;2)* 

SUG.— Apply  3>.     w  =r 1 f(a«  4-  x2)-1 

>3  --  x2    ~  2aiJ  * 


iST  +  ^  tan~'  «  +  C- 


a2  -f-  x^       2a*(a*  +  a;2)       2a3  a 


Ex.  11.  Integrate  dy  = 


(I  + 


Sen.  —  Theseformulce  often  fail,  in  consequence  of  reducing  the  expression 
to  QO,  by  making  the  denominator  0  ;  or  by  making  the  expression  indeter 
minate.  Thus  it  would  seem  at  first  glance  that  formula  £J)>  would  reduce 

y  =    I  -  —  -?  ;  but  it  will  be  found  to  fail.     Nevertheless  iheformulce  are 

J  (1—  a*)1 

of  great  practical  value  ;  and  that,  not  only  in  such  examples  as  the  above, 
which  they  reduce  directly  to  the  elementary  forms,  but  in  many  more  com 
plicated  cases  where  they  reduce  the  given  expression  to  a  form  which  can 
be  integrated  by  methods  yet  to  be  given. 


LOGARITHMIC   DIFFERENTIALS. 

103.  J*rob. — To  integrate  the  form  dy  =  X  •  log"xdx,  in  which  X 
in  an  algebraic  function  of  x. 

SOLUTION. — Put  Xdx  —  dv,  whence  lognx  =  u,   and  substitute  in  J*udv  = 
uv—fvdu.     1husy  =  fX-  lognxdx  =  log»X'  fXdx—  jT  nlog»-'zJ"(X  dx)---l. 

If  now  § ' Xdx  is  a  known  form,  we  have  made  the  integration  to  depend  upon  a 
form  in  which  the  exponent  of  logx  is  diminished.     Thus,  if  j*Xdx  =  X',  the 

X 

form  of  the  expression  to  be  integrated  becomes  —  log"— '  x  dx.     To  this  the  form 
ula  may  be  applied  again,  and  n  diminished  still  farther  if  the  algebraic  function 

X' 

— dx  can  be  integrated. 


INTEGRATION   BY   PARTS.  135 

loir  JT,  dx 
Ex.  1.  -Integrate  dy  =  — r  . 

r?r  1  _  d* 

SUG.— Put  -        -7- -  =  dv  ;  whence  log  x  —  u,  v  =  —  ^r->  and  du  -'-  ~£' 

l-l  ~r  -^)~ 
rlogxdx  _         logx^          /*      dy- 

Separating    — -  into  partial  fractions   (174\  and  integrating,  we  have 

xvl  +  x)  ~ 
rinany)y=-^^  +  log^-log(l+^)  +  ^=^logx-1°g(1  +  ^+a 

Ex.  2.  Integrate  dy  =  log  a;  (to.  y  =  ^(logo;  —  1)  +  C. 

Ex.  3.  Integrate  dy  =  x*\Q%*xdx. 

y=  ' 

Ex.  4.  Integrate  cty  = 


SOLUTION.-  Put  dv  =  ^  ;  whence  u  =  log-*  »,  v  =  log  x,  and  du  =  -  2  log-'  x-^. 

r  ^_      i  .  , 

Substituting  in  the  formula  for  integration  by  parts  y  —    I  ^^,  x)  —  log  x.  T 
.     Transposing  the  last  term  ~*-  =          «  or  ^  =  ~ 


Ex.  5.  Integrate  dy  = 


^  =  —  A  (log's  —  41og  a?  +  8)  +  C 
2 


J7  log  .77  6?J7 

Ex.  6.  Integrate  dy  =  --       —  • 
s  2 


EXPONENTIAL    DIFFERENTIALS. 

194.  Prob.—To  integrate  dy  =  xnacxdx,  when  n  is  a  positive  in 
teger. 

SOL.— Put  du  =  acxdx  ;  whence  u  =  x",  v  =  --j^a<!*>  du  =  rwrw~lda;- 
stituting  in   the  formula    fudv  =  uv  -  fvdu,   we  have    y  =  /x««"dx  = 

_J_aCT^  _    -1-  fx»-'a«dx.     Applying  the  formula  to  /x'-'a-d*,  the  inte- 
Cloga      '  «lnfffl-/ 


136  THE   INTEGRAL  CALCULUS. 

gration  is  made  to  depend  upon  the  form  Afxn-*acxdx.     Thus,  the  exponent  of  x 
can  be  finally  reduced  to  0,  and  the  integration  made  to  depend  upon  the  form 

A  fa^dx,  which  =  —  --  a"  4-  C. 
J  clog  a 


Ex.  1.  Integrate  dy  = 

eardx 


SOLUTION.  —  Put  du  =  eardx  ;   whence  u  =  a;3,   v  =  -&x,  and  du  =  3x2dx. 

a 


*.   y  =  fx3eaxdx  =  —  &x  —   -fx*ea*dx,       (repeating  the  process) 

x3          3x2         6   c 

=  —  eax eax4 — )  xeaxdx      "         "         " 

a  a2          a'iJ 

x*          3x9       ,    6x  6  r 

=   —eax gaz     I eax I   t 

a  a*       ~  a3          a?J 


a  a*        'a*          a< 

/x3       3^2   ,    6x        6\    , 
_  e<«( )  +  C. 

\Ct  CL"  Or          tt*  / 

Ex.  2.  Integrate  dy  =  x3axdx. 

dx    (  3^7*          Qx  6 

y  ___    J    ,^,3  j V.        I      /Hr 

log  a  (  log  a      log*64      logaa) 

Ex.  3.  Integrate  c??/  =  efx^dx. 


Ex.  4.  Integrate  <fi/  = 

€ 

STJG.— Put  dv  =  e~xdx,  as  before,     y  —  —  e~x(x*  +  2a  +  2)  4- 


SPECIAL   FORMS   OF   EXPONENTIALS. 


_ 

Ex.  5.  Integrate  d*/  =  —  —  -dx. 

€     --  1 


Ex.  6.  Integrate  di/  =  e^e'dx. 
SUG.— Put  exdx  =  dv,  etc.     y  =  <**  -f-  (7. 

1   4-   772 

Ex.  7.  Integrate  cfy  =  '    fdx. 

SOLUTION.  —  Put  1  -\-  x  =  z  ;  whence  x  =  z  —  1 ,  and  d#  =  dz. 

/I  _i_  >rs                 /»z«  4-  2  —  2z 
-— i-—  ercZx  =   / ez-'dz  =* 
•/ 


INTEGRATION   BY   PARTS.  137 


/0Z/7* 
—  -  ,  by  putting  do  = 

re*dz  e*         r&dz 

I  -—  =  ---  \-   I   —  . 


•  *e*dz 
and  u  =  e*,  we  have 


Substituting  this  value, 


Ex.  8.  Integrate  dy  =  ~  y  =  ~  + 


TRIGONOMETRICAL    DIFFERENTIALS. 

.  jProb.  —  To  integrate  the  forms  dy  =  sinmxdx,  dy  =  cosnxdx, 
dy  =  sinm  x  cosn  xdx. 

SOLUTION.  —To  integrate  dy  =  BinPx  dx,  put  sin  x  =  z  ;  whence  cos  x  =  (1  —  z2)  . 

dx  =  —  —  =  (1  —  z2)~2dz  ;   and  we  obtain  dy  =  zm(l  —  z2)  *dz,  which  may  be 
cos  'x 

rationalized  by  (181,  182),  or  reduced  by  one  or  more  of  formulae  g&,    JS, 

<JJ,  and  3&. 

_i 

In  like  manner  putting  cos  x  —  z,  dy  =  cos"  xdx  becomes  dy  =  —  z"(l  —  z2)  2c?z, 
and  can  be  disposed  of  as  before. 

Again,  putting  cos  x  =  z  ;  whence  sinw  x  =  (1  —  z2)  2,  cosn  x  =  z*,  and  dx  = 

---  -  =  —  (1  —  z2)  zdz,  we  have 


dt/  ==  sinm  a;  cos"  a;  da;  =  —  zn(l  •  —  z*)  2  dz  ; 

or  we  may  put  sin  x  =  z,  and  have 

*—  i 

dy  =  sin"1  x  cos"  #d#  =  zm(l  —  z2)  3  dz. 
Either  of  these  forms  may  be  treated  as  the  first. 

196.  Sen.  —  It  will  be  seen  that  this  process  will  always  effect  the  integra 
tion  when  m  and  n  are  either  positive  or  negative  integers,  and  frequently 
when  they  are  fractions.  Thus,  when  m  and  n  are  positive  even  integers, 
successive  applications  of  a&  will  reduce  the  final  integral  to  the  form 

—  &)"*dz  =  ±  I  -  —  —  =  sin"1  z,  or  cos-1^  =  x  ;  and,  when  posi- 

J  i-** 


138  THE  INTEGRAL  CALCULUS. 

tive  odd  integers,  the  final  form  will  be  ±  fz(l  —  z^ 
=F  cosx,  or  =F  sin  a?. 

When  m  and  n  are  negative  integers,  formula  ££  will  reduce  the  first  tw< 
cases. 

The  third  case  may  require  any  or  aU  of  the  four  formulae,  but  can  alwayi 
be  integrated  when  m  and  n  are  integers. 

In  many  cases  it  will  not  require  the  application  of  the  formulae,  as  wil 
be  seen  in  the  following  examples. 

Ex.  1.  Integrate  dy  =  sin3  x  dx. 

SOLUTION.  —  Putting  sin  x  =  z,  and  applying  ^.  we  have 

C  -  C  „-  oN-i  7          z^1  —  z~^  —  2/z(*  —  z^dz 

y  =  J  sm3x<ix  =  J  z3(l  —  22)    dz  =  --  ^—  -  ;  - 


=  —  ia'(l  —  z2)^  -f  1/2(1  — 


=  —  i  sin2  x  cos  x  —  J  cos  x  -f-  C. 
Ex.  2.  Integrate  dy  =  sin4  x  dx. 

y  =  —    — ^--(sin3a:  -f  f  sin  a:)  -f  fa:  -f  C. 

Ex.  3.  Integrate  dy  =  sin5  x  dx. 

y  c     \  i    13F  *    3  /    i 

Ex.  4.  Integrate  dy  =  sinfi  x  dx. 

3S 

6 


COS  X.    . 

y  = £—  (sin5  x  -\-  f  sm3  or  +  -Y-  sm  x)  -j-  -j^o;  -f  (7. 


Ex.  5.  Integrate  dy  =  cos2  x  dx. 

y  =  |(sin#cosa;  +  a7)  =  iCJ8"1^*  -f  ar)  +  (7. 
Ex.  6.  Integrate  dy  =  cos3  x  dx. 

y  =  ^  sin  #  cos2  ^  +  f  sin  a;  +  C  —  -^  sin  3x  +  |  sin  a?  -f-  (7.f 

Ex.  7.  Integrate  dy  =  cos4  x  dx. 

y  —  ^  sin  4ar  -f  ^  sin  2a;  +  f  x  -f  (7. 

Ex.  8.  Integrate  dy  =  cos5  x  dx. 


sin  5.77 
-^-+  —  - 


*  Trigonometry  (56)  sin  x  cos  a;  =  ^  sin  2z. 

t  To  effect  the  reduction    substitute   1  —  sins  x  for  cos2  x  ;    and  then  for  sins  x  substitute 
4(3  sin  a;  —  sin  3x).     (See  Trigonometry,  page  28,  Ex.  12.) 


INTEGRATION   BY   PARTS.  139 

Ex.  9.  Integrate  dy  =  cos5  x  sin5  x  dx. 

sinr^/,       sin9  a;    ,    sm4.r\ 
SUG'S.—  Putting  sin  x  =  z,  there  results  y  =  —7-^3  --  -  --  1  --  JT—  j  +  C. 

cosray          cos\r        cos'o-A 
Putting  cos  .1;  =  z,  we  nave  y  =  --  ;  —  (  3  --  -  --  1  ---  p  —  )  -f-  0  . 

QUEKY.—  What  is  the  relation  between  Caiid  0'  ? 

Ex.  10.  Integrate  dy  =  sin6  #  cos3  x  dx. 

sin2  #7 


SUG.—  If  one  factor  has  an  even  and  the  other  an  odd  exponent,  it  will  be  found 
expedient  to  put  the  function  which  has  the  even  exponent  =  z. 

Ex.  11.   Integrate  dy  =  sin4  x,  cos4xd&\ 
SOLUTION.  —  Put  sin  a;  =  z,  and  apply  <JJ. 

y  =  2       ~    °" 


Now  apply  ^  to  the  last  integral. 


Ex.  12.  Integrate  dy  == 

gUG's. — Putting  sin  x  =  z,  we  have  dy  =  z5(l  —  z2)  K< 
Applying  ^  twice  y  = (sin4  a;  +  4  sin2  as  —  8) 

O  COS  X 

Ex.  13.  Integrate  dy  =  — 


sin5  a; 
SUG'S.— Put  sin  a;  =  z,  and  we  have  dy  =  z~5(l  —  z2)~*cZz.. 


-z2)     dz,  or  restoring  x, 

_  ^LY—          3     ^  4- 1  /"-*Lf 
4    \sin4x       2  sin2  x/        *J  sin  a;' 

_  __  £o_s«/  _^ ^ — \    ,    a  log  tan  QX)  _j_   ^     (8^  ^^9^  i,) 

4    \sin-1  x       2  sin- a;/ 


140  THE  INTEGRAL  CALCULUS. 

Ex.  14.  Integrate  dy  =  —  —  . 
co&x 


sin  x/   1  4  8      \ 

___  _  /  __  i_  _  j  __  \    I    n 

5    Vcos5       3cos:}#      3cos#/ 


SUG. — Applying  £j£  three  times  the  last  term  reduces  to  0,  and  we  have  the 
result  without  integrating. 

Ex.  15.  Integrate  dy  =  — . 

sin4  x  cos2  x 

SOLUTION. — Putting  cosx  =  z,  whence  sin— 4x  =  (1  —  22)~2,  cos— 2#  =  z— 2,  and 

dx  =  —  (1  —  z2)~2dz,  we  have 
dx 


.  , 

:  ------  \-  G. 


Bestonng  x,     ---- 

cos  x  sin3  a;       3sm3a;       3  sin  x 

ANOTHEB  SOLUTION.  —  When  the  exponents  are  even,  an  elegant  solution  is  ob 
tained  by  means  of  a  special  expedient,  as  follows  : 

Introducing  the  factor  sin2  x  +  cos2  x,  which  being  equal  to  1  does  not  change 
the  value  of  the  differential,  we  have 

/dx  /•(sin2.r  +  co&x^dx  _      /*        dx  r  dx 

sin4  x  cos-  x         /         sin4  x  cos-  x  I  sin2  x  cos-  x          /   sin4  x  ~ 

/(sin2  x  -f-  cos2  x^dx          /*   dx  r   dx  r  dx  r  dx 

sin2  x  cos-  x  I   sin4  A*         /  cos'2  x          I  sin2  x          1  sin4  a;  ~~ 

cosarF    1  2    ~1    ,     _ 

—  tan  a;  —  cota;  ---    -  --  ----     +  C. 

3    Ljaan8  SB      sin  a;J 

[It  will  afford  the  student  a  good  exercise  in  trigonometrical  reduction,  to  trans 
form  the  former  expression  into  the  latter.] 


Ex.  16.  Integrate  dy  =  - 


sin  x  cos3  x 


Ex.  17.  Integrate  dy  = -dx  =  tan4  x  dx. 

COS4X 

SUG. — Put  tanx  =  z,  whence  dx  = = .     .•.  y  =  I   —dx  = 

sec-  x       1  -(-  z2  /   cos4  x 

tan4  x  dx  =  I  —  -  =  Jz^dz  —  J  dz  4-   /  =  iz3  —  z  -j-  tan—1  z  -}-  C  = 

—  tanz  +  x  4-  G 


INTEGRATION   BY  PAKTS.  141 

Ex.  18.  Integrate  dy  =  tan5  xd  x. 

y  =  ^  tan4  x  —  •£-  tan2  x  -\-  log  sec  x  -f-  0. 


197.  Prob.  —  To  integrate  the  forms  dy  =  xnsinxdx,  and 
dy  =  xn  cos  x  dx. 

SOLUTION.  —  To  integrate  dy  =  x"smxdx,  put  dv  =  siuxdx,  whence  M  =  xn, 
v  =  —cosx,  and  du  =  nxn~}dx.  .-.  y  =  —  xncosx  -f-  nfxn-*cosxdx.  By 
repeating  the  process  the  integration  will  finally  depend  upon  the  form  A  J~cos  x  dx, 


or 

In  like  manner  y  =  j*xncosxdx  =  or*  sin  x  —  nj*xn~  lsinxdx,  and  the  final 
forms  are  the  same  as  before. 

Ex.  1.  Integrate  dy  =  x*  cos  xdx. 

y  =  #3  sin  /p  _^_  3#2  cos  ^  -  (5^;  gin  #  -  Q  COS  #  +   (7. 

Ex.  2.  Integrate  d?/  =  x4  sin  or  d.r. 

24cos#-{-  (7. 


10  S.  Prop*  —  When  m  and  n  are  integers,  the  form,  dy  = 
sinm  x  cosn  x  dx  may  be  integrated  in  simple  terms  of  the  sines  and  cosines 
of  the  multiple  arcs. 

DEM.  _  The  form  sinwic  cos"#  may  be  expressed  in  simple  terms  of  multiple  area 
by  the  use  of  the  following  formulae  : 

(1)*  sin  x  sin  y  =  ±  cos  (x  —  y)—±  cos  (x  +  y\ 

(2)  cos  x  cos  y  =  i  cos  (x  —  y)  +  k  cos  (x  -f  y\ 

(3)  sin  x  cos  y  =  i  sin  (a;  +  y)  -\-  i  sin  (x  —  y). 

The  truth  of  this  statement  and  the  manner  of  applying  the  formulae,  may  be 
seen  most  readily  from  the  solution  of  a  few  examples. 

Ex.  1.  Integrate  in  simple  terms  of  the  sines  or  cosines  of  multiple 
arcs  dy  =  sin3  x  cos2  x  dx. 

SOLUTION,     sin3  x  cos2  x  =  sin  x  (sin  x  cos  jc)2 

=  sin  x[£  sin  2a;]2     [From  (3),  making  y  =  x.} 


=  ^  sin  x[^  cos  0  —  £  cos  4#]     [From  (1),  making  x  — 

y  =  2x.] 

=  £  sin  »[^  —  £  cos  4x] 
=  i  sin  as  —  \  sin  as  cos  4x 
=  $  sin  a;  —  i[£  sin  5x  —  £  sin  3x]      [From  (3),  making 

x  =  x,  and  y  =  4x.] 
=  -fc  sin  x  —  -}LQ  sin  5x  -f-  iV  sin  ^x< 

Hence  y  =  /sin"  JB  cos2  x  dx  —  ^fsinxdx  —  -h /sin  5  x  dx  -f  -^ /sin  3.7?  ate  = 
--  ^  cos  x  —  •£$  cos  3x  -f-  -gV  cos  5x  -f-  C. 

*  These  formula  are  essentially  those  of  ART.  59,  Plane  Trigonometry.    To  put  the  formula 
of  that  article  into  this  form  simply  change  x  into  £(»  +  3;)  and  y  into  £(z  —  y). 


THE   INTEGRAL   CALCULUS. 

Ex.  2.  Integrate  in  simple  terms  of  the  sines  or  cosines  of  multiple 
arcs,  dy  =  sin3  #cos3  a;  dx. 

SUG'S.     sin3  x  cos3  x  =  g  sin3  2x  =  £  sin  2x  sin2  2x  =  &  sin  2x(£  —  £  cos  4#)  = 
-j»5  sin  2^  — 
a^-  sin  6x  -f- 


.'.  y  =      sin3.'ecos3.rd.T  =  -fasantocdx  —  -^sinGxdx  -f  -aj 
—  aV  cos  2x  +  ri  2  cos  Gx  —  -$-*  cos  2x  -f-  (7. 

Ex.  3.  Integrate  in  simple  terms  of  the  sines  or  cosines  of  multiple 
arcs  dy  =  sin6  x  dx. 


SUG'S.  sin6  x  =  (sinSx)3  ~  g(l  —  cos  2.r)3  =  o  —  I  cos  2x  -f  §  cos*  2x  —  £  cos3  2x 
=  g  —  |cos2x  +  |(i  -f  £cos4x)  —  ^cos2x(i  -j-  icos4a;)  ==  -^  —  ^cos2«  + 
•ft  cos  4.r  —  -^  cos  2x  cos  4x  =  -ft  —  f0-  cos  2x  +  ft  cos  4x  —  ^  (  J  cos  2x  +  i  cos  6j) 
=  -ft  —  M  cos  2x  +  ft  cos  ^x  —  a-V  cos  6x. 

•  '  •  y  =  aV  (  -  6  sin  6x  +  |  sin  4x  —  -^  sin  2x  +  lOx)  -}-  a  [The  student  should 
be  careful  to  observe  that  the  three  formulae  given  under  the  proposition  are  suf 
ficient  to  effect  the  required  reductions.] 

Ex.  4.  Integrate  as  above  dy  =  cos3  x  dx. 

I/sin  3.37  \ 

y  =     —  -  +  3sm*-f  a 


CIRCULAR    DIFFERENTIALS. 

199*    IProb. —  To     integrate     the    forms     dy    =  f  (x)  sin"1  x  dx, 
dy  =  f  (x)  cos^x  dx,  dy  =  f(x)  tair^x  dx,  etc. 

METHOD  or  SOLUTION. — Put  f(x)dx  =  dv,  and  sin—  lx,  cos— lx,  or  tan~Jx,  as  the 
case  may  be,  =  u,  and  substitute  in  the  formula  for  integration  by  parts. 

Ex.  1.  Integrate  dy  =  x'2  sin"1  x  dx. 

SOLUTION. — Putting  x*dx  —  dv,  whence  sin—'  x  =  u,  v  —  lx3,  and  du  =  —          _. 

we  have  y  =  |x3  sin—'  x  —  \  I  —  .     Applying  formula  J^  to  the  last  inte- 

J    /I— x2 

gral  we  have  y  =  is3  sin-'  x  —  £(x2  -f-  2)-s/l  — x2  +  C. 
Ex.  2.  Integrate  dy  —  vl  —  x~  cos"1  x  dx. 

SOLUTION. — Putting  (1  —  x2)2dx  =  dv  ;  whence  u  =  cos-1  x,  v  =  lx(l  —  x°)    -f 

5  /  —  •  =  ^(1  —  ^2)?  —  £  cos—'  x,  and  du  =  —       . 

Substituting  in  the  formula  for  integration  by  parts, 

*  Ey  applying  <£J. 


INTE9RATION   BY   PARTS. 
y  =  i\x(l  —  xrf  —  cos-1  a;]  cos-1  x  — 


-f-  4x2  4-  $  J'cos-1  x  dicos-1  x) 


_  i  [a;(i  _  £2)?  _  cos-'  xlcos-1  x  4-  4x*  4-  4(cos-!  xY  -\-  C. 
=  £[x(l  —  x2)*]  cos-1  x  —  i(cos-'  xY  4-  4x2  4-  G 


Ex.  3.  Integrate  dy  =      1 

4-  (7. 

£00.    Prob.  —  ^o    integrate    the  forms    dy  =  eaxsinnxdx,    and 
dy  =  eax  cosn  x  dx. 

1 

METHOD  OF  SOLUTION.—  Put  e«*cte  =  du;   whence  sin"  a;  =  u,  v  =  -e«,  and 

du  =  n  sin"—1  x  cos  a;  dx. 

•     ni  =  —eax  sin"  x  --  •  f  eax  sin"-1  cc  cos  cc  du. 
'  J       a  aj 

Applying   the   formula   for  integration   by   parts   again,   putting  dv  =  e«rdx, 

u  =  sin"-  'a;  cos  x,    v  =  -e",  dw  =  (n  —  1)  sin"-2  .r  cos2  a;  dx  —  einnxdx,    and 

r  1 

we  have     .  •  .  y  =  J  eax  sin"  x  dx  =  -eaa;  sin"  x  — 

-  -i  -eaz  sin"-1  x  cos  a?  —  —  —  feax  sin"-2  cos2  x  dx  -}-  -feax  sin»x  dx  [  . 
a  I  a  a    J  a 

<72  +  n 
Transposing  the  last  term,  uniting  it  with  the  first,  and  dividing  by  -—  —  ,  we 

have 


sin"  x 


f 
J 


tt/ P"  /o  I      '  " 

If  now  the  last  term  be  transposed  and  united  with  the  first  member  and  we 
divide  by  the  coefficient,  we  shall  have  made  the  integration  to  depend  upon  a 
form  in  which  n  is  diminished  by  2. 

By  successive  applications  of  the  same  process,  the  integration  may  be  made  to 
depend  upon  the  form  A  (*e™dx  when  n  is  even,  or  A  far*  sin  xdx  when  n  is  odd. 
The  former  is  an  elementary  form  ;  and  the  formula  for  integration  by  parts  being 
applied  to  the  latter,  it  becomes  an  integral  without  further  process,  since  the  co 
efficient  of  the  unintegrated  term  contains  a  factor  n  —  1,  as  will  be  seen  above. 

By  a  process  altogether  similar  the  form  dy  =  e°*  cos*  a;  dx,  can  be  integrated. 

Ex.  1.  Integrate  dy  =  eax  cos  x  dx. 
N  SOLUTION. —Put  dv  =  ef^dx  ;  whence  u  =  cos  x,  v  =  -e«",  and  du  =  —  sinxd.r. 


144  THE   INTEGRAL   CALCULUS. 

y  =  Jeax  cos  x  dx  =  -e?x  cos  x  -f-  -j*eax  sin  x  dx 

=  -eax  cos  x  -\  --  eax  sin  x  --  f  a"*  cos  x  dx. 
a  '    a*  a*J 

C. 


Ex.  2.  Integrate  dy  =  e*  sin3  #  dx. 

y  =  TL<f  (sin:i  a;  +  3cos3  x  +  3  sin  #  —  6  cos  or)  -f  G. 

Ex.  3.  Integrate  dy  —  a""*  sin  for  d#. 

a  sin  for  -f  &  cos  for 
v  =  ---  :  --  1_  (7 
y  - 


£01.  Prob.—To  integrate  the  form  dy  =- r~ 

(a  -f-  bcosx)1 


/*          dx  /*(« -4- oeoBasKW  /~          a 

SOLUTION,     y  =   I  - =  I  - — !— —  =  a  /  — —— 

9         I    (a-\-bcosx)n        I   itt  4-  bcoBX)n+l          '    (a  +  6c 
•/  %/ 

5  / ! .     Applying  the  formula  for  integration  by  parts  to  the  last 

/   (a  4-  boonX)*+i 

f*       cos  x  dx                      sin  x 
integral  by  putting  cos  x  dx  =  dv,  we  have   / — —  -  = — —  -— 

/•      6  sin2  aj  dx  sin  x  x   /"(6  —  b  coss  o;)daj 

(?l  -4-  1)  I r~n  =    i —   (H  +  1)  I 

/  (a  4-  6cosa;)"+2        (a  +  6cos^<"+1  /    (a  -f- 

»/  »/ 


Substituting  this  value  in  the  preceding  we  have 
dx  r  dx  b  sin  x- 


/•         r/.i-                     /*            OZ                          osmx'  /*(?>2- — 0«COB%B 

J  (a+boosx)*         J    (a-}-bcosx/'+l~T~  (a-{-bcosx)>'+1  J   ^a_[_5COSi);) 

r  dx 

I    (a  -\-  6cosiC;w+1 


b  sin  x  f  dx 


(a  -{-  b  cos  x)"+ ' 


.  ifi  — _  fit  _i_  2rK^i  -|-  ?)cos.r"t  —  (n  4-  ?>cos.'c)2*  6  sin  a? 

(n  +  1)  /   ".TV,....  —~ dx  =  7Z- 


(a  +  beosx/ 
Transposing  and  uniting  similar  integrals, 


Dividing  by  (n  -f-  1)(62  —  a2),  and  writing  n  —  2  for  n,  we  obtain 

*  By  adding  and  subtracting  2a2  -f-  2a&  cos  a;,  62  —  62  Cos2  x  =  62  -f  2a2  -f  2a&  cos  *  —  2a2 
2ab  cos  x  —  62  cose  SB  =  62  —  as  -f  2a(a  -{-  6  cos  x)  —  (a  -j-  6  cos  x)2. 


INTEGRATION  BY  PARTS.  145 

-: __   ,  _  _*> 

i—l)(b*—a?}J  (a+b 

—  2  /•  dx 


/dx         bsinx  a(2n  —  3)       /• 

(a-\-bcosx)n       (n — l)(b* — a'^)(a-\-bcosx)n~l       (n—l)(b'2 — a2)  /  ( 

n  — 2  /•  dx 

(n  —  l)\b*  —  a*)J  (a  +  b  cos  a;;—*" 


By  means  of  repeated  applications  of  this  formula,  or  the  process  by  which 
it  was  produced,  the    integration    may  be  made    to    depend    upon   the    form 

A  C      dx 
I  a 


-\-b  cos  x 


To  integrate  dy  =  a         cog  ^  we  remember  that  cos  a;  =  cos*  - .— sin* -,  and 
cos*  ^  +  sin*  *  =  1.     Hence 

dx 


a  -|-  6  cos  x 


a(cos*  |  +  sin*  I)   -f- 


06 

ic*-da;  1         j         ^^-^  2 

XX 


i     /"      tani 

a  +  7  1  +  i^tani?  "  a 
2  *y         ra        6          2 


11       vv               ^    A         ^  *^  ^*    ~T~     ^    f       -t        •       *•*     v    .  •**  Mr    ~T~    v 

^1        •fanS  m  J-r»-*^9 

i 


r.«qpl1^5 

When  a  <  6,  we  have 


y  = 


/* 

y  (6 


(6  +  a)  — (6  — a)tan*|      ;/    (6  +  a)  —  (6- 


*__  (&_0)*  tanj 
1 


— id\^n9.)       r " 

s^i:+yj 

*\ 

>2_a2)y  (6 


tan 


xx 
2/ 


log 


,  x9  .,''*!  ,  x-r,        35 

(y2  —  a2)"  (o  -j-  tt)    —  (o  —  at)"  tan- 


146  THE  INTEG11AL  CALCULUS. 

SCH. — The  four  preceding  sections  comprise  the  greater  part  of  what  is 
known  concerning  abstract  methods  of  passing  from  the  differential  to  the 
exact  integral  function  of  a  single  variable ;  but  there  are  many  other 
methods  of  great  practical  importance  for  determining  the  approximate 
value  of  the  integral  of  a  differential  which  cannot  be  integrated  by  these 
methods.  One  of  the  most  useful  and  simple  is  given  in  the  next  section. 


SECTION   V. 

Integration  by  Infinite  Series, 

202.  It  often  occurs  that  a  differential  can  be  expanded  into  an 
infinite  series,  the  terms  of  which  can  be  integrated  separately.  If 
the  result  is  a  converging  series,  the  value  of  the  integral  may  be  de 
termined  with  sufficient  accuracy  for  practical  purposes  by  summing 
a  finite  number  of  terms.  It  may  also  sometimes  happen  that  the 
law  of  the  series  is  such  that  its  exact  sum  may  be  found,  although 
the  series  itself  is  infinite. 

This  method  is  not  only  a  last  resort  when  the  methods  of  exact 
integration  fail,  but  it  is  sometimes  serviceable  by  being  more  simple 
than  they,  even  when  they  are  applicable  ;  moreover,  it  affords  a 
method  of  developing  such  a  function. 


Ex.  1.  Integrate  dy  —  3p(l  —  x*)2dx. 


SOLUTION.—  Expanding  (1  —  x2)    by  the  binomial  or  Maclaurin's  theorem,  we 
have 

(1  _  #0*  =  1  —  $x*  —  &  —  fjoje  —  Tfgss  —  ,  etc. 


*- J*_  £iZ*'-^-^*-^+* ' 

This  series  is  converging  for  x  <  1  and  more  rapidly  converging 'as  x  is  less. 

dx 
Ex.  2.  Integrate  dy  = -. 

(l  +  *<)3 

a?*      1  -  3  x9       1  •  3  -  5  tf13  p 

y  =  a;  —  |  -  +  ^-4  ^  "  2-4-6  13  4 

Ex.  3.  Integrate  dy  =  — 


SUCCESSIVE   INTEGRATION.  147 

Ex.  4.  Integrate  dy  =  —    —  in  an  infinite  series,  and  thus  obtain  a 
1  -f-  x- 

development  of  y  =  tan""1^. 


/dx 
1+^=^--*+°=— 


(7.77 

Ex.  5.  Integrate  dy  =  —          -  in  an  infinite  series  and  thus  obtain 
\/l  —  & 

a  development  of  y  =  sin~'#. 


Ex.  6.  Integrate  dn  =  —  —  in  an  infinite  series  and  thus  obtain  a 
1  +  a? 

development  of  y  =  log  (1  +  a;). 


SECTION  VI. 

Successive  Integration, 

.  Prob.  —  To  integrate  a  second,  third,  or  nth  differential  func 
tion  of  a  single  equicrescent  variable. 

SOLUTION.  —  Since  dx  is  constant  we  may  write  J  d*y  =  J/(x)dc2  =  dxjf(x)dx, 
and  integrate  f(x)dx  as  before,  thus  reducing  the  degree  of  the  element  (differential) 
by  unity.  Putting  ff(x]dx  —  /,  (x)  +  C,  ,  we  have  fd-y  =  ff(x)d&  =  dxff(x)dx 
=  f  ,  (x)dB  +  (7,  dx.  But  J^d-y  =  di/  ;  hence  dy  =/,  (x)d»  -}-  ^i  *». 

Integrating  -again  we  have  y  =/2(«)  +  (7tx  -f  C2. 

In  like   manner  from  d*y  =f(x)dx3,  we  have  fd3y  =  d*y  =  dx^ff(x)dx  = 


Integrating  again, 

=  dy  =  dxCfl(x)dx  +  dxfC.dx  =  dx\J2(x}  +  ^o;  -f  Oe]  = 

C.xdx  + 
Integrating  a  third  time,  we  have 
£dy  =  y==  ff,(x)dx  +  (7,/. 
From  these  processes  we  deduce 


*  By  preceding  methods.  t  This  notation  signifies  the  «th  integral. 


148  THE   INTEGRAL   CALCL.LVJS. 

Ex.  1.  Integrate  d*y  —  Qa  dafl. 


SOLUTION.     fd*y  =  d*y  =  fQa  dafl  =  6adx*fdx  =  Qaxdx*  4-  (7,  *dx*. 
Again,    fd*y  =  dy  =-.  f[6axdafl  -f-  G\  dc<]  =  tiadxfxdafl  +  C}d.cfdx  = 

3axadx  +  C^xdx  -\-  C.^dx. 
Finally,  y  =  ax3  -f  i(7:x2  -f  C2x  -\-  C3. 

Ex.  2.  Integrate  d*y  —  cos  ^c  do;4. 

PROCESS.!     d3y  =  sinx  dx3  -f  <7jc7x3, 

d~y  =  —  cosxdx'2  -f-  Cjxcfcc2  -\-  C2dx*, 

-f  AC'jX^x  -f-  (72xdr  +  C.3dx, 
^  +  AC's  a*  +  6'3a;  +  6'4. 


Sen.  1.  —  It  will  be  observed  that  whatever  value  we  assign  the  constants, 
we  get  the  original  differential  by  differentiating  the  integral  as  many  times 
as  we  integrated.  The  reason  for  this,  if  not  seen  at  once,  will  appear  upon 
performing  the  operation. 

d'Ay 

Ex.  3.  Given  -~  =  0,  to  find  the  integral  y. 
CLx^ 

SOLUTION.  —  Since  the  third  differential  coefficient  is  the  differential  of  the 
second  differential  coefficient,  divided  by  the  differential  of  the  variable, 


,  n  TT  .      ,.  . 

i.  e.  —  =  ,    when   — -  =  0,  — -  must  be  a  constant     Hence  in  this  ex- 

dx*  dx  d&  dx* 

ample,  -|  =  (7,  ,  or  d*y  =  C^x^.     From  this  we  obtain  y  =  ^C^afl  +  C2x  -f  6^. 

fay        2 

F  .  4.  Given  — -  =  —  to  find  the  integral  y. 
dx3       x3 

y  =  logX  +  $Ci&  +   C2X  +   C3. 

Ex.  5.  Integrate  d*y  =  sin  x  cos2  x  dx2. 

SOLUTION. — Put  sinx  =  v  ;  whence  dv  =  cosxdx,  cos2xdx2  =  dv*,  and  d:y  = 
sin  x  cos2  x  dx'2  =  v  dv~.  From  d-y  =  v  dv~,  we  obtain  as  before  y  =  %v*  -{-  C\  v  -f-  C'2 . 
.  • .  y  =  ^-  sin3a;  -j-  (7,  sin  x  -f-  ^2- 

Ex.  6.  Integrate  dsi/  =  cos  x  sin2  a;  rfo;2. 

y  =  \  cos3  ^  +  (7j  cos  57  +  <72. 

SCH.  2. — In  order  to  integrate  successively  dny  =f(x)dxn,  according  to  the 
foregoing  process,  it  is  necessary  that  we  be  able  to  integrate  exactly  f(x)di; 
fi(.v)d:c,  f^(x}dx,  etc.  It  is  evident  that  this  will  be  frequently  impossible. 
A  method  of  approximation  which  is  often  serviceable  in  such  cases,  is 
readily  obtained  by  means  of  Maclaurin's  Formula. 

*  We  might  write  6aCi  as  the  coefficient  of  this  term  ;  but  as  Ci  represents  any  constant,  it  is 
unnecessary  to  retain  the  6a. 

t  This  is  simply  a  convenient  form  in  which  to  write  the  operation  ;  the  student  should  under 
stand  the  process  and  be  able  to  explain  it,  as  in  the  preceding  solution. 


SUCCESSIVE   INTEGRATION.  149 

204.  Prop.  _  The  nth  integral  of  f(x)dxn  may  be  developed  into  a 
series  by  the  following  formula  : 


+  cu  +  cu 


l-2-3--^n  +  L~dxJl   2-3-  -- 

,  etc 


dx* 
DEM.  —1st.   Developing  y  =  fnf(x}dxn  by  Maclaurin's  formula,  we  have 


+ete. 

-7r 


2nd.  Comparing  this  development  with  the  development  of  f"J(x)dxn  as  made 
in  the  solution  ABT.  2O1,  remembering  that  a?  =  0  in  the  bracketed  factors  of  the 
present  series,  and  that  these  factors  are  therefore  constant,  we  see  that  the  present 
series  is  that  of  AKT.  2O1  reversed,  extended  and  generalized  ;  that  [/*/(#)<&?"] 
is  C»,  [  fn~}f(x]dxn-}]  is  Cn-i,  etc.,  and  that  the  former  series  begins  with  the 

term  \_f(x)}  __  .—  __  of  the  latter.     Hence  using  <7m  CB_i,  etc.  for  these  constant 

1  •  2  •  3  ---  n 
factors  the  above  development  becomes 


y  = 


etc.     o,  E.  D. 


<  This  formula  is  readily  remembered  and  applied  by  noticing  the  law 
of  the  first  part  of  this  series  (that  containing  the  constants  C,,  Cn-i,  etc.)  ; 
and  then  observing  that  the  second  part  of  the  series  {that  from  and  includ 

ing  F  f(x\\          X"  -  1,  is  the  development  of  f(x)  by  Maclaurin's  Formula, 

o  L/  v  ;j-|_  .  2  .  3  —  n' 

each  term  being  multiplied  by  a;",  and  the   successive  terms  divided  by 
2  •  3  -  -  -  n,  3  •  4---  (n  +  1),  4  .  5-  -  -  (n  +  2)  respectively. 


Ex.    Develop    y 


r*    d.x4 
=  f*d<y==J  y=^; 


Developing  (1  —  »2)   ^  by  Maclaurin's  Formula  we  have 

*  To  differentiate  an  integral  is  to  depress  its  order,  as  will  be  seen   from  the  nature  of  th« 
processes. 


150  THE  INTEGRAL  CALCULUS. 

Hence  by  the  above  scholium,  we  obtain 

y  —  jd*y  =  /  -        -—  =  C4  -j-  Cz '-  +  Ca— \-C^ — X      -f-      x  . 

l-3a-8  '  1.3-53! 


' 


SECTION    VII. 

Definite  Integration  and  the  Constants  of  Integration, 

205.  DEF. — An  Indefinite  Integral  is  one  in  which  the 
constant  or  constants  of  integration  remain  undetermined,  and  which 
has  not  been  satisfied  by  any  particular  value  of  the  variable. 

ILL. — All  the  integrals  hitherto  produced  are  indefinite  integrals. 

200.  DEF. — A  Corrected  Integral  is  one  in  which  the  value 
of  the  constant  (or  constants)  of  integration  has  been  determined 
and  substituted  for  the  general  symbol  C. 

20 Y.  DEF. — An  integral  is  said  to  be  taken  between  Limits, 
when  the  indefinite  integral  has  been  satisfied  for  two  different  values 
of  the  variable,  and  the  difference  between  these  results  taken. 

208.  DEF. — A  Definite  Integral  is  an  integral  taken  be 
tween  limits. 

A  definite  integral  and  the  limits  between  which  it  is  taken  is  sym 
bolized  thus  :  y  =•  I  f(.x)dx.  This  signifies  that  the  indefinite  inte 
gral  otf(x)dx  is  to  be  obtained,  and  first  satisfied  by  substituting  b 
for  xt  then  by  substituting  a  for  x ;  and  that,  finally,  the  latter  is  to  be 
subtracted  from  the  former,  x  =  a  and  x  =  b  are  called  the  limits 
of  the  integral,  the  former  being  called  the  inferior  and  the  latter  the 
superior  limit.* 

Ex.  1.  Find  the  value  of  y  =    /     5x2dx. 

•A 

SUG'S. — The  indefinite  integral  is  y'  =  f.r3  -f-  C. 

Now  this  is  true  for  all  values  of  x,  hence  for  x  =  3.    For  this  value  of  x,  we  have 

*  It  is  assumed  that  x  can  have  such  values  as  we  assign  it,  arid  that  the  function  is  continuous 
between  these  values,  t.  e.  that  it  does  not  become  imaginary  or  infinite  for  any  intermediate 
value  of  x. 


DEFINITE  INTEGRATION  AND  THE  CONSTANTS  OF  INTEGRATION.    151 

y"    =  45  -f-  C.  In  like  manner  for  x  =  1 

we  have  y'"  =    £ -f-  £  Subtracting  the  latter 

/3 
5x-dx  =  43£. 

Ex.  2.  Find  the  definite  integral  of  dy  —  nxdx  between  the  limits 

/"'                    n(b*  —  a2) 
a  and  b.  y  =    /     n^tfj;  = . 

Jo,  A 

Ex.  3.  Find  the  value  ofy=j     (x*dx  • —  b2x  dx).     Atis.,  — ^64. 


209.   Disposing  of   the   Constant  of  Integration. 

There  are  two  principal  methods  of  disposing  of  the  constant  of  in 
tegration,  C  : 

1st.  By  integrating  between  limits,  the  constant  is  eliminated.  This 
is  illustrated  in  the  preceding  examples. 

2nd.  "When  there  is  anything  in  the  nature  of  the  problem  under 
discussion,  from  which  we  can  know  the  value  of  the  function  for 
some  particular  value  of  the  variable,  by  substituting  these  values  in 
the  indefinite  integral,  the  value  of  the  constant  C  can  be  found. 
And,  as-  C  is  a  constant,  if  we  find  its  value  for  any  particular  value 
of  the  variable,  it  has  the  same  value  for  all  values  of  the  variable. 

Ex.  1.     Find    the    corrected     integral    of     the    function    dz    = 
t  _j_  %ax  dx2)*,  on  the  hypothesis  that  z  =  0  when  x  =  0. 


8  £ 

SUG'S.  —  The  indefinite  integral  is  z  =  -7r-(l  +  *axf  +  &     Now  if  z  =  0  when 

O  Q 

x  __  o?  we  have  0  =  -  ---  f-  C.     .  •  .    C  =  —  —  .     Substituting  this  value  of  C  in 

O  i  Q 

the  indefinite  integral,  we  have,  as  the  corrected  integral,  z  =•—-(!  -j-  \ax)'z  —  —  -. 
Ex.  2.   What  is  the  value  of  C,  when  du  =  -  —  (p*  +  y~)2dy,  if 


u  =  0  when  y  =  0  ?  Ans.t  C=  — 

Ex.  3.  Given  du  =  (2r)i(2r'—  y)~^dyt  what  is  the  value  of  G,  if 
u  =  0  when  y  =  0  ?  "What  if  u  =  0  when  y  =  2r  ? 

Answers,  4r,  0. 

TT/T..  —  A  differential  is  one  of  the  infinitesimal  elements  of  which  a  quantity  is 
conceived  as  composed.  Thus,  let  A  represent  the  area  of  the  surface  lying  be 
tween  AM  and  AX,  Fig.  35;  PD  being  any  ordinate,  and  ab  the  consecutive 


152 


THE  INTEGRAL   CALCULUS. 


ordinate,  PDab  may  be  considered  as  representing  an 
element  of  the  area,  or  dA.  This  element  in  the  case 
of  the  parabola  is  found  to  be  \/^px  dx ;  hence  d  A  = 

\/2/>x  dx.  This  expression  therefore  represents  any  one 
of  the  infinitesimal  elements  of  which  the  quantity  A 
is  composed. 

The  Indefinite  Integral  \B  A  =  |(%>)  *    4-  C. 

This  is  indefinite  in  two  respects.     1st,  it  is  true  for  any 
value  of  x  ;  2nd,  it  is  indefinite,  and  in  fact  indetermi-  FIG.  35. 

nate,  as  regards  C,  the  value  of  which  may  be  anything. 

As  REGARDS  THE  CONSTANT,  if  ice  choose  to  estimate  the  area  from  A,   so  that 
A  =  0  when  x  =  0,  we  have  0  =  0  -[-  (7.     .*.    (7=0.     Hence  the  corrected  integral 

is  A  =  3(2p)5x2.  This  represents  the  area,  estimated  from  A  to  any  value  we  may 
choose  to  give  x.  If  x  =  A  D ',  A  =  area  A  P  D '.  Again,  if  we  choose  to  estimate 
the  area  from  the  focal  ordinate  H  F,  calling  A  =  0,  when  x  =  ?,p,  we  have 

0  =  K%>)  ($p)    4~  ^>  or  %P~  4"  &     .'.   C  —  —  ip-,  and  the  corrected  integral  is 

A  =  l((lpfxz  —  ip5.  This  represents  the  area  estimated  from  the  focal  ordinate 
H  F  to  any  value  we  choose  to  give  x.  Thus  if  x—  AD',  this  corrected  integral 
represents  the  area  H  FP'  D'.  Finally,  suppose  we  ask,  Where  must  the  area  be 
conceived  as  beginning  in  order  that  C  =  —  m  ?  To  meet  this  case  we  have 

o  =  fflbA 


m,  whence  x  =    *j  — .     If  therefore  the  area  is  conceived  as  com- 

A.     /  »Xv\ 


mencing  at  the  ordinate  corresponding  to  x==  3  I—,  C=  —  m.     Thus  we  perceive 

the  indeterminate  character  of  C.  We  also  observe  the  limits  of  its  possible 
values.  In  this  case  C  may  be  any  negative  quantity,  but  no  positive  quantity. 

'  INTEGRATION  BETWEEN  LIMITS  is  illustrated  by  considering  the  area  as  estimated 
from  some  possible  place  (no  matter  where)  and  extending  1st  to  x  =  a,  whence 

A'  =  (2p;2a2  4-  (7;  and  2nd  to  x  =  b,  whence  A"  =  (2p)  b'1  4-  C.  Now  the  dif 
ference  between  A'  and  A"  will  represent  the  area  between  the  ordinates  corres 
ponding  to  x  =  a  and  x  =  b.  Call  this  A'"',  and  A'"  =  (2p)  (b2  —  «2 ),  if  a  << b. 

Letting  AD  =  a  and  AD'  =b,  A'"  =  (2p)  (b*  —  a*)  =  area  PDP'D'. 

This  subject  will  have  more  ample  illustration  in  Sections  IX. — XIII.  inclusive, 
which  the  student  is  now  prepared  to  read. 


THE  END. 


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